Full text of "Opticks: or, A treatise of the reflections, refractions, inflexions and colours of light : also two treatises of the species and magnitude of curvilinear figures"

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The Dibner Library
of the History of
Science and Technology

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ADVERTISEMENT.

P Art of the enfuing Difcourfe about Light was written
at the defire of fome Gentlemen of the Royal Society,
in the Tear 1675. and then fent to their Secretary, and
read at their Meetings, and the reft was added about
Twelve Tears after to complete the Theory; except the
Third Book, and the lajt Proportion of the Second, which
wereJince put together out offcattered Papers. To avoid
being engaged in Difputes about thefe Matters, I have
hitherto delayed the Printing, and jhould fill have de¬
layed it, had not the importunity of Friends prevailed upon
me. If any other Papers writ on . this Suh\eB are got out
of my Hands they are imperfeff, and were perhaps written
before / had tried all the Experiments here fet down ,
and fully fatisfied my felf about the Laws of Ref rations
and Compofition of Colours. I have here Publifhed what
I think proper to come abroad, wijhing that it may not be
Tranflated into another Language without my Confent.

The Crowns of Colours, which fometimes appear about
the Sun and Moon, I have endeavoured to give an Ac¬
count of ; but for want of fufficient Obfervations leave that
Matter to be further examined. 'The Subject of the Third
Book I have alfo left imperfeffi, not having tried all the

Expe-

Experiment# which I intended when J was about thefe
Matters, nor repeated fome of thofe which I did try, until
I hadJ'atisfied my felf about all their Circumftances. To
communicate what I have tried, and leave the resit to
others for further Enquiry, is all my Dejign in publifhing
thefe Tapers.

In a Letter written to Mr.Leibnitz in the Tear 167 6 .
and pubhfhed by Dr. Wallis, I mentioned a Method by
which I had found fome general Theorems about fquaring
Curvilinear Figures, or comparing them with the Conic
Sections, or other the Jimple ft Figures with which they may
he compared. And fome Tears ago I lent out a Manufcript
containing fuch Theorems, and havingJince met with fome
Things copied out of it, / have on this Occajim made it
publick, prefixing to it an Introdudlion and fubjoyning a
Scholium concerning that Method. And I have pined
with it another fmall Trail concerning the Curvilinear
Figures of the Second, R ind, which was alfo written,
many Tears ago, and made known to fome Friends, who
have foMdied the making it publick ; ~

- - 4

I N.

DS!

The FIRST BOOK

O F

O P T I C K S.

PART I.

M Y Defign in this Book is not to explain the Pro¬
perties of Light by Hypothefes, but to propofe
and prove them by Reafon and Experiments:
In order to which , I fhall premife the following Defini¬
tions and Axioms.

DEFINITIONS.

D E F I N. I.

B Y the (Rays of Light I under ft and its leaf Tarts, and thofe
as wellSucceffiVe in the fame Lines as Contemporary in fe¬
deral Lines . For it is manifeft that Light confifts of parts
both Succeflive and Contemporary 5 becaufe in the fame
place you may flop that which comes one moment, and
let pals that which comes prefently after5 and in the fame
time you may flop it in any one place, and let it pafs in
any other. For that part of Light which is ftopt cannot
be the fame with that which is let pafs. The leaft Light
or part of Light, which may be ftopt alone without the
reft of the Light, or propagated alone, or do or fuffer any

A thing

0 3

thing alone, which the reft of the Light doth not or fuft
ers not, I call a Ray of Light.

D E F I N. IT.

^efrangihility of the Bays of Light, is their Difpofition to he
refra&ed or turned out of their Way in pafjing out of one tranfi
parent Body or Medium into another . And a greater or lefs Bp-
frangihility of Bays, is their Difpofition to he turned more or lefs
out of their Way in like Incidences on the fame Medium . Mathe¬
maticians ufually confider the Rays of Light to be Lines
reaching from the luminous Body to the body illumina¬
ted, and the refraCtion of thofe Rays to be the bending
or breaking of thofe Lines in their paffing out of one Me¬
dium into another. And thus may Rays and Refractions
be confidered, if Light be propagated in an inftant. But
by an Argument taken from the ./Equations of the times
of the Eclipfes of Jupiter’s Satellites it feems that Light is
propagated in time, (pending in its pafiage from the Sun
to us about Seven Minutes of time : And therefore I have
chofen to define Rays and Refractions in fuch general
terms as may agree to Light in both cates.

DEFIN. III.

Bpfiexihility of Bays, ts their Difpofition to he turned hac^ into
the fame Medium from any other Medium upon whofe Surface they
fall. And Bays are more or left refiexihle , which are returned
hack^ more or lefs eafily. As if Light pafs out of Glafs into
Air, and by being inclined more and more to the conrr-
mon Surface of the Glafs and Air, begins at length to be
totally reflected by that Surface 3 thofe forts of Rays which
at like Incidences are reflected moft copioufly, or by in¬
clining the Rays begin fooneft to be totally reflected, are
moft reflexible, D E-

E $ ]

DEFIN. IV.

The Angle of Incidence , is that Angle which the Line defcribed
by the incident Pay contains with the (perpendicular to the refle¬
cting or refraMing Surface at the Point of Incidence .

DEFIN. V.

The Angle of Pyflexion or Pyfruition, is the Angle which the
Line defcribed by the refleBed or ref railed Pay containeth with
the Perpendicular to the reflecting or refralling Surface at the
Point of Incidence .

D EFI N. VI.

The Sines of Incidence, Reflexion, and pefraltion, are the
Sines of the Angles of Incidence, peflexion, and Pefraltion.

DEFIN. VII.

The Light whofe Pays are all alike PyfTangible, I call Sim¬
ple , Homogeneal and Similar $ and that whofe Pays are flome
more Pyfrangible than others, l call Compound, Heterogene at and
Dijfimilar. The former Light I call Homogeneal, not
becaufe I would affirm it fo in all relpe&s $ but becaufe
the Rays which agree in Refrangibility 5 agree at leaft in
all thofe their other Properties. Which I confider in the
following Difcourfe.

DEFIN. VUE

The Colours of Homogeneal Lights y I call Primary, Homo¬
geneal and Simple y and thofe of Heterogeneal Lights, Heteroge-
neal and Compound . For thefe are always compounded of
the colours of Homogeneal Lights 5 as will appear in the
following Difcourfe. A z " A XI-

C 4 1

AXIOMS.

AX. I.

T HE Angles of Incidence, Inflexion, and RefraSion, lye
in one and the fame Plane.

AX. II.

The Angle of Reflexion is equal to the Angle of Incidence „

A X. III.

If the refraBed (Ray he returned direSly hack L to the Point
of Incidence , it Jhall he ref railed into the Line before defer L
led by the incidmt Ray.

AX. IV.

RefraBion out of the rarer Medium into the denfer , is made
towards the Perpendicular 3 that is, fo that the Angle of Refra -
Sion he left than the Angle of Incidence.

A X. V.

The Sine of Incidence, is either accurately1 or Very ?iearly in a
gtyen Ratio to the Sine of Refra&ion.

Whence if that Proportion be known in~ any one Incli¬
nation of the incident Ray, *tis known in all the Inclina^
tions, and thereby the Refra&ion in all cafes of Incidence
on the fame refra&ing Body may be determined. Thus
if the Refra&ion be made out of Air into Water, the Sine
©f Incidence of the red Light is to the Sine of its Refra¬
ction as 4 to 3 . If out of Air into Glafs, the Sines are
’ * as

as 17 to it. In Light of other Colours the Sines have
other Proportions : but the difference is fo little that it
need feldom be confidered.

Suppofe therefore, that R S reprefents the Surface of Fig. i
ftagnating Water, and C is the point of Incidence in-
which any Ray coming in the Air from A in the Line
A C is reflected or refracted, and I would know whether
this Ray {hall go after Reflexion or Refraction : I ereCt
upon the Surface of the Water from the point of Inci¬
dence the Perpendicular C P and produce it downwards
to Q., and conclude by the firfl: Axiom, that the Ray af¬
ter Reflexion and Refraction, fhall be found fomewhere in
the Plane of the Angle of Incidence A C P produced. I
let fall therefore upon the Perpendicular C P the Sine of
Incidence A D, and if the reflected Ray be defired, I pro¬
duce A D to B fo that D B be equal to A D, and draw
C B. For this Line C B fhall be the reflected Ray ; the
Angle of Reflexion B C P and its Sine B D being 7 equal
to the Angle and Sine of Incidence, as they ought to be
by the fecond Axiom. But if the refraCted Ray be de¬
fired, I produce A D to H, fo that D H may be to A B
as the Sine of Refraction to the Sine of Incidence; that is
as 3 to 4 ; and about the Center C and in the Plane AGP
with the Radius C A defcribing a Circle ABE I draw
Parallel to the Perpendicular GPQ, the Line H E cutting
the circumference in E, and joyning CE, this Line CE •

(hall be the Line of the refraCted Ray. For if E F be let
fall perpendicularly on the Line P Q., this Line E F fhall,
be the Sine of Refraction of the Ray C E, the Angle of
Refraction being ECQ ; and this Sine EF is eoual to

DH, and confequently in Proportion to the Sine of Inci-
dence. AD as 3 to.4.,

I11

Fix . z .

<_>

-[ 6 ]

In like manner, if there be a Prifm of Glafs (that is a
Glafs bounded with two Equal and Parallel Triangular
ends, .and three plane and well polifhed Sides, which meet
in three Parallel Lines running from the three Angles of
one end to the three Angles of the other end) and if the
Refradion of the Light in paffing crofs this Prifm be defi-
red : Let ACB reprefent a Plane cutting this Prifm tranf-
verfly to its three Parallel lines or edges there where the
Light pafleth through it, and let JE be the Ray inci¬
dent upon the firft fide of the Prifm A C where the Light
goes into the Glafs 5 And by putting the Proportion of
the Sine of Incidence to the Sine of Refradion as 17 to
11 find E F the firft refraded Ray. Then taking this Ray
for the Incident Ray upon the fecond fide of the Glafs B C
where the Light goes out, find the next refraded Ray F G
by putting the Proportion of the Sine of Incidence to the
Sine of Refradion as 11 to 17. For if the Sine of Inci¬
dence out of Air into Glafs be to the Sine of Refradion
as 17 to 11, the Sine of Incidence out of Glafs into Air
muft on the contrary be to the Sine of Refradion as 11
to 17, by the third Axiom.

Much after the fame manner , if A C B D reprefent a
Glafs {pherically Convex on both fides (ufually called a
Lem, fuch as is a Burning-glafs, or Spectacle-glafs, or an
Objed-glafs of a Telefcope) and it be required to know
how Light falling upon it from any lucid point Q {hall
be refracted, let Q_M reprefent a Ray falling upon any
point M of its firft fpherical Surface A CB, and by ereding
a Perpendicular to the Glafs at the point M, find the firft
refracted Ray M N by the Proportion of the Sines 17
to 11. Let that Ray in going out of the Glafs be inci¬
dent upon N, and then find the fecond .refracted Ray N q
by the Proportion of the Sines 11 to 17. And after the

C 7 J

feme manner may the Refraction be found when the
Lens is Convex on one fide and Plane or Concave on
the other, or Concave on both Sides.

AX. VI.

Homogeneal (Rays which flow from federal (points of any Qk~
and fall almofi Perpendicularly on any reflecting or refra*
Bing Plane or Spherical Surface , flhall afterwards diverge from
fo many other Points , or he Parallel to fo many other Lines , or
converge to fo many other Points , either accurately or without any
fenjible Error. And the fame thing will happen , if the pays he
refleBed or refraBed fuccejfiVely by two or three or more Plane
or flherical Surfaces.

The Point from which Rays diverge or to which they
converge may be called their Focus. And the Focus of
the incident Rays being given, that of the reflected or re¬
fracted ones may be found by finding the Refraction of
any two Rays, as above y or more readily thus.

Caf. i. Let ACBbe a reflecting or refracting Plane, Fg
and Q the Focus of the incident Rays, and Qf C a per¬
pendicular to that Plane. And if this perpendicular be
produced to q, fo that q C be equal to QC, the point q
fhall be the Focus of the reflected Rays. Or if q C be
taken on the fame fide of the Plane with QC and in Pro¬
portion to QC as the Sine of Incidence to the Sine of

Refra&ion, the point q fhall be the Focus of the refrac¬
ted Rays.

Caf. 2 Let A C B be the reflecting Surface of any Fk
Sphere whofe Center is E. Bifect any Radius thereof (fup-
pole EC) in T, and if in that Radius on the fame fide the
point T you take the Points Q and q, fo that T Q, T E,
and Tq be continual Proportionals, and the point Qbe

the:

[ 8 ]

the Focus of the incident Rays, the point q (hall be the
Focus of the refleded ones.

Caj\ 4. Let ACB be the refracting Surface of any
Sphere whofe Center is E, In any Radius thereof E C
produced both ways take E T and C t feverally in fuch
Proportion to that Radius as the leffer of the Sines of
Incidence and Refradion hath to the difference of thofe
Sines. And then if in the fame Line you find any two
Points Q_ and q , fo that T Q be to E T as E t to t q y
taking t q the contrary way from t which T Q. lieth from
T, and if the Point Q.be the Focus of any incident Rays,
the Point q fhall be the Focus of the refracted ones.

And by the fame means the Focus of the Rays after
two or more Reflexions or Refradions may be found.

Caf. 4. Let A C B D be any refrading Lens, fpheri-
cally Convex or Concave or Plane on either fide, and let
CD be its Axis (that is the Line which cuts both its Sur¬
faces perpendicularly, and paffes through the Centers of
the Spheres,) and in this Axis let F and/be the Foci of the
refraded Rays found as above, when the incident Rays
on both fides the Lens are Parallel to the fame Axis 5 and
upon the Diameter F / bifected in E, defcribe a Circle.
Suppofe now that any Point Q. be the Focus of any inci¬
dent Rays. Draw QE cutting the faid Circle in T and t J
and therein take t q in fuch Proportion to t E as t E or T E
hath to T Q_. Let t q lye the contrary way from t which
T Q. doth from T, and q fhall be the Focus of the refrac¬
ted Rays without any fenfible Error , provided the Point
Q_ be not fo remote from the Axis, nor the Lens fo broad
as to make any of the Rays fall too obliquely on the
refracting Surfaces.

And by the like Operations may the reflecting or re¬
fracting Surfaces be found when the two Foci are given,

-- " : • and

[ 9 ]

and thereby a Lens be formed, which fhall make the Rays
flow towards or from what place you pleafe.

So then the meaning of this Axiom is, that if Rays
fall upon any Plane or Spherical Surface or Lens, and
before their Incidence flow from or towards any Point <X,
they fhall after Reflexion or Refraction flow from or to¬
wards the Point q found by the foregoing Rules. And if
the incident Rays flow from or towards ieveral points Q.,
the reflected or refracted Rays fhall flow from or towards
fo many other Points q found by the fame Rules. Whe¬
ther the reflected and refracted Rays flow from or towards
the Point q is eafily known by the fituation of that Point.

For if that Point be on the fame fide of the reflecting or
refracting Surface or Lens with the Point Q, and the in¬
cident Rays flow 7 from the Point Q, the reflected flow to¬
wards the Point q ai id the refracted from it 5 and if the
incident Rays flow towards Q, the reflected flow from q 7
and the refracted towards it. And the contrary happens
when q is on the other fide of that Surface.

A X. VII.

Wherever the pays which come from all the (points of any Ob-
jefl meet again in fo many Points after they hair been made to
converge hy Reflexion or Pyfraftion, there they mil make a Pic¬
ture of the Object upon any white Body on ivhich they fall

So if PR reprefent any Object without Doors, and ABPg. 3.
be a Lens placed at a hole in the Window-flhut of a dark
Chamber, whereby the Ray s that come from any Point Q.
of that Object are made to converge and meet again in
the Point q ^ and if a Sheet of white Paper be held at q
for the Light there to fall upon it : the Picture of that
Object P R will appear upon the Paper in its proper Shape '

B ^ ~ and

[ IO !

and Colours. For as the Light which comes from the
Point Q goes to the Point q, fo the Light which comes
from other Points P and R of the Object, will go to fo
many other correfpondent Points p and r (as is manifeft
by the flxth Axiom 3) fo that every Point of the Object
{hall illuminate a correfpondent Point of the Picture, and
thereby make a Picture like the Object in Shape and Co¬
lour, this only excepted that the Picture ihall be inverted.
And this is the reafon of that Vulgar Experiment of call¬
ing the Species of Objects from abroad upon a Wall or
Sheet of white Paper in a dark Room.

In like manner when a Man views any Object P QR 5
the Light which comes from the feveral Points of the Ob¬
ject is fo refracted by the tranfparent skins and humours
of the Eye, (that is by the outward coat EFG called the
Tunica Cornea , and by the cryftalline humour AB which is
beyond the Pupil m 4 J as to converge and meet again at
fo many Points in the bottom of the Eye, and there to paint
the Picture of the Object upon that skin (called the T«-
nica Retina) with which the bottom of the Eye is covered.
For Anatomills when they have taken off from the bot¬
tom of the Eye that outward and moll thick Coat called
the Dura Mater , can then fee through the thinner Coats
the Pictures of Objects lively painted thereon. And thefe
Pictures propagated by Motion along the Fibres of the Op-
tick Nerves into the Brain, are the caufe of Vifion, For
accordingly as thefe Pictures are perfect or imperfect, the
Object is feen perfectly or imperfectly. If the Eye be tin¬
ged with any colour ("as in the Difeafe of the Jaundife) fo
as to tinge the Pictures in the bottom of the Eye with that
Colour, then all Objects appear tinged with the fame Co¬
lour. If the humours of the Eye by old Age decay, fo
as. by Ihrinking to make the Carma and Coat of the Cry -

ftalline

[II ]

flatting humour grow flatter than before, the Light will not be
refracted enough, and for want of a fufficient RefraCtion
will not converge to the bottom of the Eye but to feme
place beyond it, and by confequence paint in the bottom
of the Eye a confufed Picture,and according to the indiftinCt-
nefs of this Picture the ObjeCt will appear confufed. This
is the reafon of the decay of Sight in old Men, and flie.ws
why their Sight is mended by Spectacles. For thofe Con¬
vex-glafles fupply the defeCt of plumpnefs in the Eye, and
by encreafing the RefraCtion make theRays converge fooner
fo as to convene diftinCtly at the bottom of the Eye if the
Glafs have a due degree of convexity. And the contrary
happens in fhort-fighted Men whofe Eyes are too plump.

For the RefraCtion being now too great,the Rays converge
and convene in the Eyes before they come at the bottom *
and therefore thePiCture made in the bottom and the Vifion
caufed thereby will not be diftinCt, unlefs the ObjeCt be
brought fo near the Eye as that the place where the con¬
verging Rays convene may be removed to the bottom, or
that the plumpnefs of the Eye be taken off and the Refra¬
ctions diminimed by a Concave-glafs of a due degree of
Concavity, or laftiy that by Age the Eye grow flatter till it
come to a due Figure : For fliort-fighted Men fee remote
Objects belt in Old Age, and therefore they are accounted
to have the moft lafting Eyes.

A X. VIII.

An Object feen by Reflexion or fraction^ appears in that place
from ivhence the 5 ^ays after their laft ^flexion or RefraCtion di¬
verge in falling on the Spectators Eye .

If the ObjeCt A be feen by Reflexion of a Looking- Fig
glafs m b 7 it (hall appear, not in it’s proper place A, but

B i behind

[ 12 ]

behind the Glafs at a, from whence any Rays AB, AC ?
A D, which flow from one and the fame Point of the Ob-
jed, do after their Reflexion made in the Points B, C, D,
diverge in going from the Glafs to E, F, G, where they
are incident on the Spectator’s Eyes. For thefe Rays do
make the fame Picture in the bottom of the Eyes as if
they had come from the Object really placed at a without
the interpolation of the Looking-glafs 3 and all Villon is
made according to the place and lhape of that Picture.

In like manner the Object D feen through a Prifm ap¬
pears not in its proper place D, but is thence tranflated to
fome other place d lituated in the laft refracted Ray F G
drawn backward from F to L

And fo the Object Q_ feen through the Lens A B, appears
at the place q from whence the Rays diverge in paffing
from the Lens to the Eye. Now it is to be noted, that the
Image of the Object at q is fo much bigger or lefler than
the Object it felfat Q, as the diftance of the Image at
q from the Lens AB is bigger or lefs than the diftance of
the Object at Q_ from the fame Lens. And if the Object
be feen through two or more fuch Convex or Concave-
glafles, every Glafs lhall make a new Image, and the Ob-
jed fliall appear in the place and of the bignefs of the laft
Image. Which confideration unfolds the Theory of Mi-
crolcopes and Telefcopes. Eor that Theory confifts in al-
moft nothing elfe than the defcribing fuch Glafles as fhall
make the laft Image of any Object as diftind and large
and luminous as it can conveniently be made.

I have now given, in Axioms and their Explications the
flimm of what hath hitherto been treated of in Opticks*
For what hath been generally agreed on I content my
felf to afltime under the notion oT Principles, in order to
what I have further to write. And this may fuffice for an

Intro-

Introduction to Readers of quick Wit and good Under^
ftanding not yet verfed in Opticks : Although thofe who
are already acquainted with this Science, and have
handled Glaffes, will more readily apprehend what fol¬
lowed!.

PROPOSITIONS.

I P1^0 ( P. I. Theor. I.

L IGHTS which differ in Colour, differ alfo in De^
grees of Refrangibility.

The Proof by Experiments:

Exper. i. I took a black oblong ftiff Paper terminated
by Parallel Sides, and with a Perpendicular right Line
drawn crofs from one Side to the other , diftinguiflbed it
into two equal Parts.. One of thefe Parts I painted with
a red Colour and the other with a blew. The Paper was
very black, and the Colours intenfe and thickly laid on,
that the Phenomenon might be more confpicuous. This
Paper I viewed through a Prifm offolid Glaft, whofe two
Sides through which the Light paffed to the Eye were
plane and well polifhed, and contained an Angle of about
Sixty Degrees : which Angle I call the refrading Angle of
the Prifm. And whilft I viewed it , I held it before a
Window in fuch manner that the Sides of the Paper were
parallel to the Prifm, and both thofe Sides and the Prifm
parallel to the Horizon, and the crofs Line perpendicular
to it 3 and that the Light which fell from the Window

[ H ]

■upon the Paper made an Angle with the Paper, equal to
that Angle which was made with the fame Paper by the
Light renewed from it to the Eye. Beyond the Prifm was
the Wall of the Chamber under the Window covered over
with black Cloth, and the Cloth was involved in Dark-
nefs that no Light might be reflected from thence, which
in paffing by the edges of the Paper to the Eye , might
mingle it felf with the Light of the Paper and obfeure the
Phenomenon thereof. Thefe things being thus ordered,
I found that if the refracting Angle of the Priim be turned
upwards, fo that the Paper may feem to be lifted upwards
by the Refraction, its blew half will be lifted higher by
the RefraCtion than its red half. But if the refraCting
Angle of the Prifm be turned downward, fo that the Pa¬
per may feem to be carried lower by the RefraCtion, its
blew half will be carried fomething lower thereby than
its red half. Wherefore in both cafes the Light which
comes from the blew half of the Paper through the Prifm
to the Eye, does in like Circumftances fuffer a greater Re¬
fraction than the Light which comes from the red half,
and by confequenee is more refrangible.

llhifiration. In the Eleventh Figure, M N represents the
Window,and D E the Paper terminated with parallel Sides
D J and H E, and by the tranfverfe Line F G diftinguiflied
into two half's, the one D G of an intenfely blew Colour,
the other FEof an intenfely red. AndBACc#^ repre-
fents the Prifm whole refraCting Planes AV>b a and AC ca
meet in the edge of the refraCting Angle A 4. This edge
A a being upward, is parallel both to the Horizon and to
the parallel edges of the Paper D J and H E. And de re-
prelents the Image of the Paper feen by RefraCtion up¬
wards in fuch manner that the blew half D G is carried

hiaher to dg than the red half F E is to fe y and therefore
D fuffers

[ 15 ]

fuffers a greater Refradion. If the edge of the refrading;
Angle be turned downward, the Image of the Paper will
be refraded downward fuppofe to «N, and the blew half
will be refraded lower to ^ y than the red half is to
Exper. 2 . About the aforefaid Paper, whofe two halfs
were painted oyer with red and blew, and which was fluff
like thin Paftboard, I lapped feveral times a flender thred
of very black Silk, in fuch manner that the feveral parts ;
of the thred might appear upon the Colours like fo many
black Lines drawn over them, or like long and flender
dark Shadows caft upon them. I might have drawn black
Lines with a Pen, but the threds were fmaller and better
defined. This Paper thus coloured and lined I fet againft
a Wall perpendicularly to the Horizon, fo that one of the
Colours might Hand to the right hand and the other to
the left. Clofe before the Paper at the confine of the Co¬
lours below I placed a Candle to illuminate the Paper
ftrongly : For the Experiment was tried in the Night*
The flame of the Candle reached up to the lower edge of
the Paper, or a very little higher. Then at the diftance of
Six Feet and one or two Inches from the Paper upon the
Floor I ereded a glafs Lens four Inches and a cjuarter
broad, which might colled the Rays coming from the
feveral Points of the Paper, and make them converge to¬
wards fo many other Points at the fame diftance of fix
Feet and one or two Inches on the other fide of the Lens,
and fo form tne Image of tne coloured Paper upon a white
Paper placed there 5 after the fame manner that a Lens at
a hole in a Window cafts the Images of Objeds abroad i
upon a Sheet of white Paper in a dark Room. The afore¬
faid white Paper, ereded perpendicular to the Horizon
and to the Rays which fell upon it from the Lens, I moved!
fometimes towards-the Lens, fomerimes from it, to find 1

thee

tltf]

the places where the Images of the blew and red parts of
the coloured Paper appeared mod diftind. Thofe places
I eafily knew by the Images of the black Lines which I
had made by winding the Silk about the Paper. For the
Images of thofe fine and (lender Lines (which by reafon of
their blacknefs were like Shadows on the Colours) were
^onfufed and fcarce vifible, unlefs when the Colours on ei¬
ther fide of each Line were terminated mod diftindly.
Noting therefore, as diligently as I could, the places where
the Images of the red and blew halfs of the coloured Pa¬
per appeared mod diftind, I found that where the red
half of the Paper appeared diftind, the blew half appeared
eonfufed, fo that the black Lines drawn upon it could
fcarce be feen 5 and on the contrary where the blew half
appeared mod diftind the red half appeared eonfufed, fo
that the black Lines upon it were fcarce vifible. And be¬
tween the two places where thefe Images appeared diftind
there was the diftance of an Inch and a half : the diftance
of the white Paper from the Lens, when the Image of the
red half of the coloured Paper appeared mod diftind, be-
ing greater by an Inch and an half than the diftance of the
fame white Paper from the Lens when the Image of the
blew half appeared moft diftind. In like Incidences there¬
fore of the blew and red upon the Lens, the blew was re-
fraded more by the Lens than the red, fo as to converge
fooner by an Inch and an half, and therefore is more refran¬
gible.

. Illuftratio?i. In the Twelfth Figure, DE fignifies the co¬
loured Paper, D G the blew half, F E the red half, M N
the Lens, H J the white Paper in that place where the red
half with its black Lines appeared diftind, and hi the fame
Paper in that place where the blew half appeared diftind.
The place hi was nearer to the Lens MN than the place
H J by an Inch and an half. Scholium .

[

Scholium. The fame things fiicceed notwithstanding that
fome of the Circumftances be varied : as in the firft Ex¬
periment when the Prifm and Paper are any ways inclined
to the Horizon, and in both when coloured Lines are
drawn upon very black Paper. But in the Defcription of
thefe Experiments , I have fet down fuch Circumftances
by which either the Phenomenon might be rendred more
conlpicuous, or a Novice might more eafily try them, or
by which I did cry them only. The fame thing I have
often done in the following Experiments : Concerning all
which this one Admonition may fuffice. Now from thefe
Experiments it follows not that all the Light of the blew
is more Refrangible than all the Light of the red 3 For
both Lights are mixed of Rays differently Refrangible,
So that in the red there are fome Rays not lefs Refrangible
than thofe of the blew , and in the blew there are fome
Rays not more Refrangible than thofe of the red 3 But
thefe Rays in Proportion to the whole Light are but few,
and ferve to diminifh the Event of the Experiment, but
are not able to deftroy it. For if the red and blew Co¬
lours were more dilute and weak, the diftance of the Ima¬
ges would be lefs than an Inch and an half 3 and if they
were more intenfe and full, that diftance would be greater,
as will appear hereafter. Thefe Experiments may fuffice
for the Colours of Natural Bodies. For in the Colours
made by the Refradtion of Prifms this Propofition will
appear by the Experiments which are now to follow in the
next Propofition.

C tp^O (P.

\ ■

P R 0 P. II. Theor. II.

The Light of the Sun confifls of Rays differently Refrangible,

The Proof by Experiments.

er.g. I N a very dark Chamber at a round hole about
j[ one third part of an Inch broad made in the
Shut of a Window I placed a Glafs Prifm, whereby the
beam of the Suns Light which came in at that hole might
be refracted upwards toward the oppofite Wall of the
Chamber, and there form a coloured Image of the
Sun.- The Axis of the Prifm (that is the Line paffing.
through the middle of the Prifm from one end of it to
the other end Parallel to the edge of the Refracting Angle)
was in this and the following Experiments perpendicular
to the incident Rays. About this Axis I turned the Prifm
flowly , and faw the refraCted Light on the Wall or co¬
loured Image of the Sun firft to defcend and then to af-
cend. Between the Defcent and Afcent when the Image
feemed Stationary 5 I ftopt the Prifm, and fixt it in that
Pofture, that it flhould be moved no more. For in that
pofture the Refractions of the Light at the two fides of
the RefraCting Angle, that is at the entrance of the Rays
into the Prifm and at their going out of it, were equal to
one another. So alfo in other Experiments as often as I
would have the Refractions on both fides the Prifm to be
equal to one another, I noted the place where the Image
of the Sun formed by the refraCted Light flood ftill be¬
tween its two contrary Motions, in the common Period
of its progrefs and egrefs 3 and when the Image fell upon
that place, I made fail the Prifm. And in this pofture, as

the

[ J 9]

the moft convenient,it is to be underftood that all the Prifms
are placed in the following Experiments, unlefs where feme
other pofture is defcribed. The Prifm therefore being pla¬
ced in this pofture, I let the refraded Light fall perpendi¬
cularly upon a Sheet of white Paper at the oppofite Wall
of the Chamber, and obferyed the Figure and Dimenfions
of the Solar Image formed on the Paper by that Light.
This Image was Oblong and not Oval, but terminated
with two Redilinear and Parallel Sides, and two Semi¬
circular Ends. On its Sides it was bounded pretty diftindly,
but on its Ends very confufedly and indiftindly, the Light
there decaying and vanifhing by degrees. The breadth qf
this Image anfwered to the Suns Diameter, and was about
two Inches and the eighth part of an Inch , including the
Penumbra. For the Image was eighteen Feet and an half
diftant from the Prifm, and at this diftance that breadth if
diminilhed by the Diameter of the hole in the Window-lhut,
that is by a quarter of an Inch, febtended an Angle at the
Prifm of about half a Degree, which is the Sun’s apparent
Diameter. But the length of the Image was about ten Inches
and a quarter, and the length of the Rectilinear Sides about
eight Inches 3 And the refrading Angle of the Prifm where¬
by fo great a length was made, was -6a degr. With a Ids
Angle the length of the Image was lefs, the breadth re¬
maining the fame. If the Prifm was turned about its Axis
that way which made the Rays emerge more obliquely out
of the fecond refolding Surface of the Prifm, the Image foon
became an Inch or two longer, or more 3 and if the Prifm
was turned about the contrary way, fo as to make the Rays
fall more obliquely on the firft refrading Surface, the Image
foon became an Inch or two (hotter. And therefore in try¬
ing this Experiment, I was as curious as I could be in pla¬
cing the Prifm by the above-mentioned Rule exadiy in

C i fuch

[ 20 ]

fuch a pofture that the Refractions of the Rays at their emer¬
gence out of the Prifm might be equal to that at their inci¬
dence on it. This Prifm had feme Veins running along
within the Glafs from one end to the other, which feat-
tered feme of the Sun’s Light irregularly, but had no fen-
fible effeCt in encreafing the length of the coloured Spec¬
trum. For I tried the fame Experiment with other Prifms
with the fame Succefs. And particularly with a Prifm
which feemed free from fuch Veins, and whofe refracting
Angle was 62- Degrees, I found the length of the Image 92

or 10 Inches at the diftance of i8 J Feet from the Prifm,
the breadth of the hole in the Window-flhut being i of an

Inch as before. And becaufpit is eafie to commit a mi-
ftake in placing the Prifm in its due pofture, I repeated
the Experiment four or five times, and always found the
length of the Image that which is fet down above. With
another Prifm of clearer Glafs and better Polliflh, which
feemed free from Veins and whofe refracting Angle was
63 Degrees, the length of this Image at the fame diftance

of 18; Feet was alfo about 10 Inches, or 10k Beyond

tkefe Meafures for about - 1 or 1 of an Inch at either end of

4 3

the SpeCtrum the Light of the Clouds feemed to be a little
tinged with red and violet, but fo very faintly that I fufpe-
Cted that tinClure might either wholly or in great meafure
arife from feme Rays of the SpeCtrum fcattered irre¬
gularly by feme inequalities in the Subftance and Polifh
of the Glafs, and therefore I did not include it in thefe
Meafures. Now the different Magnitude of the hole in
theWindow-fliut, and different thicknefs of the Prifm where
the Rays paffed through it, and different inclinations of the
Prifm to the Florizon, made no fenfible changes in the
length of the Image. Neither did the different matter of

the

[21 ]

the Prifms make any : for in a Veffel made of polifihedr
Plates of Glafs cemented together in the Ihape of a Prifmn
and filled with Water, there is the like Succefs of the Ex¬
periment according to the quantity of the Refraction. It
is further to be obferved, that the Rays went on in right
Lines from the Prifm to the Image, and therefore at their
very going out of the Prifm had all that Inclination to
one another from which the length of the Image pro^
ceeded, that is the Inclination of more than two Degrees
and an half. And yet according to the Laws of Opticks
vulgarly received, they could not poflibly be fo much in¬
clined to one another. For let EG reprefent the Window-
fliut, F the hole made therein through which a beam of the
Sun’s Light was tranfmitted into the darkned Chamber, and
ABC a Triangular Imaginary Plane whereby the Prifm is
feigned to be cut tranfverfly through the middle of the
Light. Or if you pleafe, let AB C reprefent the Prifm it
felfj looking direCtly towards the Spectator s Eye with its
nearer end : And let X Y be the Sun, M N the Paper upon
which the Solar Image or Spectrum is call, and P T the
Image it felf whofe fides towards V and W are Rectili¬
near and Parallel, and ends towards P and T Semicir¬
cular. YKHP and X L J T are the two Rays, the firft
of which comes from the lower part of the Sun to the
higher part of the Image, and is refraCted in the Prifm at
K and H, and the latter comes from the higher part of
the Sun to the lower part of the Image, and is refraCted
at L and J. Since the Refractions on both fides the Prifm
are equal to one another, that is the RefraCtion at K equal
to the RefraCtion at J, and the RefraCtion at L equal to
the RefraCtion at H, fo that the Refractions of the inci¬
dent Rays at K and L taken together are equal to the
RefraCtions. of the. emergent Rays at Hand J taken toge¬
ther t:

i>2]

thef : it follows by adding equal things to equal things,
that the RefraCtions at K and H taken together, are equal
to the Refractions at J and L taken together , and there¬
fore the two Rays being equally refracted have the fame
Inclination to one another after RefraCiion which they had
before, that is the Inclination of half a Degree anfwering
to the Sun's Diameter. For fo great was the Inclination
of the Rays to one another before RefraCHon. So then,
the length of the Image P T would by the Rules of Vul¬
gar O-pticks fubtend an Angle of half a Degree at the
Prifm, and by confequence be equal to the breadth Vw $
and therefore the Image would be round. Thus it would
be were the two Rays X L J T and Y K H P and all the
reft which form the Image P w T >, alike Refrangible.
And therefore feeing by Experience it is found that the
Image is not round but about five times longer than
broad, the Rays which going to the upper end P of the
Image fuffer the greateft Refraction, muft be more Refran¬
gible than thofe which go to the lower end T , unlefs the
inequality of Refraction be cafual.

This Image or SpeCtrum P T was coloured, being red
at its lead: refracted end T, and violet at its mod: refraCted
end P, and yellow green and blew in the intermediate
fpaces. Which agrees with the firft Fropofition, that Lights
which differ in Colour do alfo differ in Refrangibility.
The length of the Image in the foregoing Experiments I
meafured from the fainted; and outmoft red at one end, to
the fainted: and outmoft blew at the other end.

Exper. 4. In the Sun's beam which was propagated in¬
to the Room through the hole in the Window-fhut, at
the diftance of feme Feet from the hole, I held the Prifin
in fech a pofture that its Axis might be perpendicular to
that beam. Then I looked through the Prifm upon the

[ 23 ]

hole, and turning the Prifm to and fro about its Axis to
make the Image of the hole afcend and defcend, when be¬
tween its two contrary Motions it feemed ftationary, X
ftopt the Prifm that the Refractions on both fides of the
refracting Angle might be equal to each other as in the
former Experiment* In this Situation of the Prifm view¬
ing through it the faid hole, I obferved the length of its
refracted Image to be many times greater than its breadth,
and that the mod refracted part thereof appeared violet,
the leaf: refracted red, the middle parts blew green and
yellow in order. The fame thing happened when I re¬
moved the Prifm out of the Suns Light, and looked
through it upon the hole fhining by the Light of the
Clouds beyond it. And yet if the Refraction were done
regularly according to one certain Proportion of the Sines
of Incidence and Refraction as is vulgarly fuppofed, the
refracted Image ought to have appeared round.

So then, by thefe two Experiments it appears that m
equal Incidences there is a confiderable inequality of Re¬
fractions : But whence this inequality arifes, whether it be
that feme of the incident Rays are refracted more and--,
others lefs, conftantly or by chance, or that one and the
fame Ray is by^ Refraction difturbed, flattered, dilated,
and as it were Iplit and Ipread into many diverging Rays,,,
as Grimaldo fuppofes, does not yet appear by thefe Experi¬
ments, but will appear by thofe that follow.

Exper. 5. Confidering therefore, that if in the third Ex¬
periment the Image of the Sun fheuld be drawn out into-
an oolong form, either by a Dilatation of every Ray, or
by any other cafual inequality of the Refractions, the fame,
oblong Image would by a fecond Refraction made Side¬
ways be drawn out as much in breadth by the like Dila¬
tation. of the. Rays or other cafual inequality of the Re¬
fractions.

I 2 4]

fractions Sideways, I tried what would be the Effects of
fuch a fecond Refraction. For this end I ordered all things
as in the third Experiment, and then placed a fecond Prifm
immediately after the firft in a crofs Petition to it, that it
might again refrad the beam of the Suns Light which
came to it through the firft Prifm. In the firft Prifm this
beam was refraded upwards, and in the fecond Sideways.
And I found that by the Refradion of the fecond Prifm
the breadth of the Image was not increafed, but its fupe-
rior part which in the firft Prifm buffered the greater Re¬
fradion and appeared violet and blew, did again in the
fecond Prifm buffer a greater Refradion than its inferior
part, which appeared red and yellow , and this without
any Dilation of the Image in breadth.

4- llluflratton. Let S reprefent the Sun, F the hole in the
Window, ABC the firft Prifm, D H the fecond Prifm, Y
the round Image of the Sun made by a dired beam of
Light when the Prifms are taken away, P T the oblong
Image of the Sun made by that beam palling through the
•firft Prifm alone when the fecond Prifm is taken away, and
ft the Image made by the crofs Refradions of both
Prifms together. Now if the Rays which tend towards
the feveral Points of the round Image Y were dilated and
Ipread by the Refradion of the firft Prifm, fo that they
ftiould not any longer go in fingle Lines to Angle Points,
but that every Ray being fplit, fhattered, and changed
from a Linear Ray to a Superficies of Rays diverging
from the Point of Refradion, and lying in the Plane of
the Angles of Incidence and Refradion, they Ihould
go in thofe Planes to fo many Lines reaching almoft
from one end of the Image P T to the other, and if
that Image Ihould thence become oblong : thofe Rays
and their feveral parts tending towards the feveral Points of

[^ 5 ]

the Image P T ought to be again dilated and fpread Side¬
ways by the tranfverfe RefraCtion of the fecond Prifm, fo
as to compofe a fourfquare Image, fuch as is reprefented
at «r 7 . For the better underftanding of which, let the Image
PT be diftinguiftied into five equal Parts PQK, KQRL,
LRSM, MSVN, NVT. And by the fame irregularity
that the Orbicular Light Y is by the RefraCtion of the firft
Prifm dilated and drawn out into a long Image P T, the
the Light PQK which takes up a fpace of the fame length
and breadth with the Light Y ought to be by the Refra¬
ction of the fecond Prifm dilated and drawn out into the
long Image *qkp, and the Light K QR L into the long
Image 4 q r h an ^ ^ Lights LRSM, MSVN, NVT
into fo many other long Images l r s w, m s v w, ny tl 3 and
all thefe long Images would compofe the fourfquare Image
* 1 - Thus it ought to be were every Ray dilated by Re¬
fraction, and fpread into a triangular Superficies of Rays
diverging from the Point of RefraCtion. For the fecond
RefraCtion would fpread the Rays one way as much as the
firft doth another, and fo dilate the Image in breadth as
much as the firft doth in length. And the fame thing
ought to happen, were fome Rays cafually refraCted more
than others. But the Event is otherwife. The Ima^e P T
was not made broader by the RefraCtion of the Ifecond
Prifm, but only became oblique, as ? tis reprefented at p
its tipper end P being by the RefraCtion tranflated to a
greater diftance than its low 7 er end T. So then the Light
which went towards the upper end P of the Image, was
(at equal Incidences) more refraCted in the fecond Prifm
than the Light which tended towards the lower end T,
that is the blew and violet, than the red and yellow 3 and
therefore was more Refrangible. The fame Light was by
the RefraCtion of the firft Prifm tranflated further from the

[2(5]

place Y to which it tended before Refraction 5 and there¬
fore fuffered as well in the firft Prifm as in the fecond a
greater Refraction than the reft of the Light, and by con-
lequence was more Refrangible than the reft, even before
its incidence on the firft Prifm.

Sometimes I placed a third Prifm after the fecond, and
foitietimes alfo a fourth after the third , by all which the
Image might be often refracted fide ways : but the Rays
which were more refracted than the reft in the firft Prifin
were alfo more refracted in all the reft, and that without
any Dilatation of the Image fide ways : and therefore thofe
Rays for their conftancy of a greater Refraction are de-
fervedly reputed more Refrangible.

But that the meaning of this Experiment may more
clearly appear, it is to be confidered that the Rays which
are equally Refrangible do fall upon a circle anfwering to
the Suns Difque. For this was proved in the third Experi¬
ment. By a circle I underftand not here a perfect Geo¬
metrical Circle, but any Orbicular Figure whofe length is
equal to its breadth, and which, as to fenfe, may feem
circular. Let therefore A G reprefent the circle which all
the moft Refrangible Rays propagated from the whole
Difq ue of the Sun, would illuminate and paint upon the
oppofite Wall if they were alone 5 E L the circle which all
the leaft Refrangible Rays would in like manner illuminate
and paint if they were alone 3 B H, C J, D K, the circles
which fo many intermediate forts of Rays would fuccef-
fively paint upon the Wall, if they were fingly propagated
from the Sun in fucceffive Order, the reft being always in¬
tercepted 3 And conceive that there are other intermediate
Circles without number which innumerable other inter¬
mediate forts of Rays would fucceffively paint upon the
Wall if the Sun fhould fucceffively emit every fort apart.

[ 27 ]

And feeing the Sun emits all thefe forts at once, they muft
all together illuminate and paint innumerable equal cir¬
cles, of all which, being according to their degrees of Re-
frangibility placed in order in a continual feries, that ob¬
long Spedrum PT is compofed which I defcribed in the
third Experiment. Now if the Sun's circular Image Y
which is made by an unrefraded beam of Light was by
any dilatation of the fingle Rays, or by any other irregu¬
larity in the Refraction of the firft Prifm, converted into
the Oblong Spedrum, P T : then ought every circle A G,
B H, C J, tsrc. in that Spedrum, by the crofs Refra¬
ction of the fecond Prifm again dilating or otherwife
fcattering the Rays as before, to be in like manner drawn
out and transformed into an Oblong Figure, and thereby
the breadth of the Image P T would be now as much aug¬
mented as the length of the Image Y was before by the Re¬
traction of the firft Prifm 5 and thus by the Refradions of
both Prifms together would be formed a fourfqnare Figure
T*t 7 as I defcribed above. Wherefore fince the breadth of
the SpeCtrum P T is not iiicreafed by the Refraction fide¬
ways, it is certain that the Rays are not fplit or dilated, or
otherways irregularly fcattered by that RefraCtion, but
that every circle is by a regular and uniform RefraCtion
tranflated entire into another place, as the circle A G by
the greateft RefraCtion into the place ag y the circle B H by
a Ids RefraCtion into the place bh, the circle C J by a Re¬
fraCtion frill lefs into the place cz, and fo of the reft 5 by
which means a new Spedrum p t inclined to the former
P T is in like manner compofed of circles lying in a
right Line 3 and thefe circles muft be of the fame bignefs
with the former, becaufe the breadths of all the Spe-

drums Y, PT and pt at equal diftances from the Prifms
are equal.

D 2

I con-

[28]

t confidered further that by the breadth of the hole F
through which the Light enters into the Dark Chamber*
there is a Penumbra made in the circuit of the SpeCtrum
Y, and that Penumbra remains in the rectilinear Sides of
the SpeCtrums P T and pt . I placed therefore at that hole
a Lens or ObjeCt-glafs of a Telefcope which might call
the Image of the Sun diftinCtly on Y without any Penum-
bra at all, and found that the Penumbra of the Rectili¬
near Sides of the oblong SpeCtrums P T and pt was alfo
thereby taken away, fo that thofe Sides appeared as di¬
ftinCtly defined as did the Circumference of the firft Image
Y. Thus it happens if the Glafs of the Prifms be free
from veins, atd their Sides be accurately plane and well
polifihed without thofe numberlefs waves or curies which
ufually arife from Sand-holes a little fmoothed in polifh-
ing with Putty. If the Glafs be only well polifbed and
free from veins and the Sides not accurately plane but a
little Convex or Concave, as it frequently happens 3 yet
may the three SpeCtrums Y, P T and pt want Penumbras,
but not in equal diftances from the Prifms. Now from
this want of Penumbras, I knew more certainly that every
one of the circles was refraCted according to fome moft
regular, uniform, and conftant law. For if there were
any irregularity in the RefraCtion, the right Lines A E and
G L which all the circles in the SpeCtrum P T do touch,
could not by that RefraCtion be tranflated into the Lines
a e and gl as diftinCt and ftraight as they were before, but
there would arife in thofe tranflated Lines fome Penumbra
or crookednefs or undulation, or other fenfible Perturba¬
tion contrary to what is found by Experience. Whatfo-
ever Penumbra or Perturbation fbould be made in the
circles by the crofs RefraCtion of the fecond Prifm, all
that Penumbra or Perturbation would be conlpicuous in

[ 29 ]

the right Lines a e and g l which touch thofe circles. And
therefore fince there is no fuch Penumbra or Perturbation
in thofe right Lines there mu ft be none in the circles*
Since the diftance between thofe Tangents or breadth of
the Sped rum is not increafed by the Refradions, the Dia¬
meters of the circles are not increafed thereby. Since thofe
Tangents continue to be right Lines, every circle which
in the firft Prifm is more or lefs refraded , is exadly in
the fame Proportion more or lefs refraded in the fecond.
And feeing all thefe things continue to fucceed after the
fame manner when the Rays are again in a third Prifm,
and again in a fourth refraded Sideways, it is evident that
the Rays of one and the fame circle as to their degree of
Refrangibility continue always Uniform and Homogeneal
to one another, and that thofe of feveral circles do differ
in degree of Refrangibility, and that in fome certain and
conftant Proportion. Which is the thing I was to prove.

There is yet another Circumftance or two of this Ex- Fig.
periment by which it becomes ftill more plain and con¬
vincing. Let the fecond Prifm D H be placed not imme-
ately after after the firft, but at fome diftance from it 5
Suppofe in the mid-way between it and the Wall on which
the oblong Spedrum P T is caft, fo that the Light from
the firft Prifm may fall upon it in the form of an oblong
Spedrum, *7 Parallel to this fecond Prifm,and be refraded
Sideways to form the oblong Spedrum p t upon the Wall.
And you will find as before, that this Spedrum p t is in¬
clined to that Spedrum P TL, which the firft Prifm forms
alone without the fecond 3 the blew ends P and p beino^ fur¬
ther diftant from one another than the red ones T and t
and by confluence that the Rays which go to the blew
end *■ of the Image *1 and which therefore fuffer the greateft
Refradion in the firft Prifm, are again in the fecond Prifm
more refraded than the reft,

13 °]

The fame thing I try’d alfo by letting the Sun’s Light
into a dark Room through two little round holes F and p
made in the Window, and with two Parallel Prifins ABC
and plaoed at thofe holes ( one at each ) refra&ing
thofe two beams of Light to the oppofite Wall of the
Chamber, in fuch manner that the two colour’d Images
P T and m n which they there painted were joyned end to
end and lay in one ftraight Line, the red end T of the
one touching the blew end m of the other. For if thefe
two refra&ed beams were again by a third Prifm D H pla¬
ced croft to the two firft, refrafted Sideways, and the Spe-
Ctrums thereby tranflated to feme other part of the Wall
of the Chamber, fuppofe the Spectrum P T to p t and
the Spedxum M N to m w, thefe tranflated SpeCtrums p t
and m n would not lie in one ftraight Line with their ends
contiguous as before, but be broken off from one another
and become Parallel, the blew end of the Image m n being
by a greater Refraction tranflated farther from its former
place M T, than the red end t of the other Image p t from
the fame place MT which puts the Propofition paft di-
fpute. And this happens whether the third Prifm D H be
placed immediately after the two firft or at a great diftance
from them , fo that the Light refraCfed in the two firft
Prifms be either white and circular, or coloured and ob¬
long when it falls on the third.

Exper. 6 . In the middle of two thin Boards I made
round holes a third part of an Inch in Diameter, and in
the Window-fhut a much broader hole, being made to let
into my darkned Chamber a large beam of the Sun’s
Light 3 I placed a Prifm behind the Shut in that beam to
refraCl it towards the oppofite Wall, and clofe behind the
Prifm I fixed one of the Boards, in fuch manner that the

middle of the refraCted Light might pafs through the hole

made

[30

made in it, and the reft be intercepted by the Board.
Then at the diftance of about twelve Feet from the firft
Board I fixed the other Board, in fuch manner that the
middle of the refracted Light which came through the hole
in the firft Board and fell upon the oppofite Wall might
pafs through the hole in this other Board, and the reft be-r
ing intercepted by the Board might paint upon it the co*
loured SpeCtrum of the Sun. And clofe behind this Board
I fixed another Prifm to refrad: the Light which came
through the hole. Then I returned Ipeedily to the firft
Prifm, and by turning it flowly to and fro about its Axis,

I caufed the Image which fell upon the fecond Board to
move up and down upon that Board, that all its parts
might fucceflively pafs through the hole in that Board and
fall upon the Prilm behind it. And in the mean time, I
noted the places on the opppfite Wall to which that Light
after its Refraction in the fecond Prifm did pafs $ and by
the difference of the places I found that the Light which
being moft refraCted in the firft Prifm did go to the blew
end of the Image, was again more refrafted in the fecond
Prifm than the Light which went to the red end of that
Image, which proves as well the firft Propofition as the
fecond. And this happened whether the Axis of the two
Prifms were parallel, or inclined to one another and to the
Horizon in any given Angles.

Illuftration. Let F be the wide hole in the Window-ftiut,
through which the Sun ftiines upon the firft Prifm ABC, ^
and let the refra&ed Light fall upon the middle of the
Board D E, and the middle part of that Light upon the
hole G made in the middle of that Board. Let this tra-
jeCted part of the Light fall again upon the middle of the
fecond Board d e and there paint fuch an oblong coloured
Image of the Sun as was defcribed in the third Experiment.

[ 32 ]

By turning the Prifm ABC flowly to and fro about its
Axis this Image will be made to move up and down the
Board d e, and by this means all its parts from one end to
the other may be made to pafs fucceffively through the
hole g which is made in the middle of that Board. In the
mean while another Prifm ale is to be fixed next after
that hole£ to refradt the trajedted Light a fecond time.
And thefe things being thus ordered, I marked the places
M and N of the oppofite Wail upon which the refradted
Light fell,and found that whilft the two Boards and fecond
Prifm remained unmoved, thofe places by turning the firft
Prifm about its Axis were changed perpetually. For when
the lower part of the Light which fell upon the fecond
Board d e was call through the hole g it went to a lower
place M on the Wall, and when the higher part of that
Light was call through the fame hole^, it went to a higher
place N on the Wall, and when any intermediate part of
the Light was call through that hole it went to fome place
on the Wall between M and N. The unchanged Pofition
of the holes in the Boards, made the Incidence of the Rays
upon the fecond Prifm to be the fame in all cafes. And
yet in that common Incidence fome of the Rays were more
refradted and others lefs. And thofe were more refracted
in this Prifm which by a greater Refradtion in the firft
Prifm were more turned out of the way, and therefore for
their conftancy of being more refradted are defervedly cal¬
led more Refrangible.

Exper . 7. At two holes made near one another in my
Window-fhut I placed two Prifms , one at each, which
might caft upon th§ oppofite Wall (after the manner of
the third Experiment) two oblong coloured Images of the
Sun. And at a little diftance from the Wall I placed a
long (lender Paper with ftraight and parallel edges, and

ordered

[ 33 1

ordered the Prifms and Paper fo, that the red Colour of
one Image might fall direddy upon one half of the Paper,
and the violet colour of the other Image upon the other
half of the fame Papery fo that the Paper appeared of two
Colours , red and violet, much after the manner of the
painted Paper in the firft and fecond Experiments. Then
with a black Cloth I covered the Wall behind the Paper,
that no Light might be reflected from it to difturb the
Experiment, and viewing the Paper through a third Prifm
held parallel to it, I faw that half of it which was illumi¬
nated by the Violet-light to be divided from the other
half by a greater Refradion, efpecially when I went a good
way off from the Paper. For when I viewed it too near
at hand, the two halfs of the Paper did not appear fully
divided from one another, but Termed contiguous at one
of their Angles like the painted Paper in the firft: Expe¬
riment. Which alfo happened when the Paper w r as too
broad.

Sometimes inftead of the Paper I ufecl a white Thred,
and this appeared through the Prifm divided into two Pa¬
rallel Threds as is reprelented in the i pth Figure , where Fig.
D G denotes the Thred illuminated with violet Light
from D to E and with red Light from E to G, and d e °fg
are the parts of the Thred feen by Refraction. If one half
or the 1 hred be conftantly illuminated with red, and the
other half be illuminated with all the Colours lucceffively,
(which may be done by caufihg one of the Prifms to be
turned about its Axis whilft the other remains unmoved)
tins other half in viewing the Thred through the Prifm,
will appear in a continued right Line with the firft half
when illuminated with red, and begin to be a little divi¬
ded from it when illuminated with Orange, and remove
further from it when illuminated with Yellow, and ftill

E further

[ 34 ]

further when with Green, and further when with Blew, and
go yet further off when illuminated with Indigo, and fur-
theft when with deep Violet. Which plainly flhews, that
the Lights of feveral Colours are more and more Refran¬
gible one than another, in this order of their Colours, Red,
Orange, Yellow, Green, Blew, Indigo, deep Violet 3 and
fo proves as well the firft Propofition as the fecond.

Fig-. 17. I caufed alfo the coloured SpeCtrums P T and M N
made in a dark Chamber by the Refractions of two Prifrns
to lye in a right Line end to end, as was defcribed above
in the fifth Experiment, and viewing them through a third
Priftn held Parallel to their length, they appeared no longer
in a right Line, but became broken from one another, as
they are reprefented at p t and m n\ the violet end m of the
SpeCtrum m n being by a greater RefraCtion tranflated
further from its former place M T than the red end £ of the
other SpeCtrum pt.

_f/V. 20. I further caufed thofe two SpeCtrums P T and MN to
become co-incident in an inverted order of their Colours,
the red end of each falling on the violet end of the other,
as they are reprefented in the oblong Figure P T M N 3
and then viewing them through a Prifm D H held Paral¬
lel to their length, they appeared not co-incident as when
viewed with the naked Eye, but in the form of two di-
ftinCt SpeCtrums p t and mn crofting one another in the
middle after the manner of the letter X. Which fhews
that the red of the one SpeCtrum and violet of the other,
which were co-incident at P N and M T, being parted
from one another by a greater RefraCtion of the violet to
p and m than of the red to n and t, do differ in degrees of
Refrangibility.

I illuminated alfo a little circular piece of white Paper
all over with the Lights of both Prifms intermixed, and

when

when it was illuminated with the red of one Spedhrum and
deep violet of the other , fo as by the mixture of thofe
Colours to appear all over purple , I viewed the Paper,
firft at a lefs diftance , and then at a greater , through a
third Prifm 3 and as I went from the Paper, the refradted
Image thereof became more and more divided by the un¬
equal Refradfion of the two mixed Colours, and at length
parted into two diftindt Images, a red one and a violet one,
whereof the violet was furtheft from the Paper, and there¬
fore buffered the greateft Refradtion. And when that Prifm
at the Window which call: the violet on the Paper was ta¬
ken away,the violet Image difappeared} but when the other
Prifm was taken away the red vanifhed : which (hews that
thefe two Images were nothing elfe than the Lights of the
two Prifms which had been intermixed on the purple Pa¬
per, but were parted again by their unequal Refradtions
made in the third Prifm through which the Paper was
viewed. This alfo was obfervable that if one of the
Prifms at the Window, fuppofe that which caft the violet y
on the Paper, was turned about its Axis to make all the
Colours in this order, Violet, Indigo, Blew, Green, Yel¬
low, Orange, Red, fall fucceffively on the Paper from that
Prifm, the violet Image changed Colour accordingly, and
in changing Colour came nearer to the red one, until when
it was alfo red they both became fully co-incident.

I placed alfo two paper circles very near one another,
the one in the red Light of one Prifm, and the other in
the violet Light of the other. The circles were each of
them an Inch in Diameter, and behind them the Wall was
dark that the Experiment might not be difturbed by any
Light coming from thence. Thefe circles thus illuminated,

I viewed through a Prifm fo held that the Refradtion might
be made towards the red circle , and as I went from them

E 2 they

they came nearer and nearer together, and at length be¬
came co-incident 3 and afterwards when I went ftill further
off, they parted again in a contrary order, the violet by a
greater Refraction being carried beyond the red.

Exper. 8. In Summer when the Stiffs Light ufes to
be ftrongeft, I placed a Prifrn at the hole of the Window-
flint, as in the third Experiment, yet fo that its Axis might
be Parallel to the Axis of the World, and at the oppofite
Wall in the Stiffs refra&ed Light, I placed an open Book.
Then going Six Feet and two Inches from the Book, I
placed there the abovementioned Lens,by which the Light
reftedled from the Book might be made to converge and
meet again at the diftance of fix Feet and two Inches be¬
hind the Lens , and there paint the Species of the Book
upon a fiheet of white Paper much after the manner of the
iecond Experiment. The Book and Lens being made faft,
1 noted the place where the Paper was, when the Letters
of the Book, illuminated by the fulled: red Light of the
Solar Image falling upon it, did caff their Species on that
Paper moft diftinctly • And then I flay'd till by the Mo¬
tion of the Sun and confequent Motion of his Image on
the Book, all the Colours from that red to the middle of
the blew pafs’d over thofe Letter and when thofe Letters
were illuminated by that blew, I noted again the place of
the Paper when they caff their Species moft diftindtly upon
it : And I found that this laft place of the Paper was nearer
to the Lens than its former place by about two Inches and
an half, or two and three quarters. So much fooner there¬
fore did the Light in the violet end of the Image by a grea¬
ter Refraction converge and meet, than the Light in the
red end. But in trying this the Chamber was as dark as I
could make it. For if thefe Colours be diluted and weak-
ned by the mixture of any adventitious Light, the diftance

between.

between the places of the Paper will not be fo great. This
diftance in the fecond Experiment where the Colours of
natural Bodies were made ufe of, was but an Inch and a
half, by reafon of the imperfedtion of thofe Colours. Elere
in the Colours of the Prifm , which are manifeftly more
full, intenfe, and lively than thofe of natural Bodies, the
diftance is two Inches and three quarters. And were the
Colours ftill more full, I queftion not but that the di¬
ftance would be considerably greater. For the coloured
Light of the Priim, by the interfering of the Circles de~
fcnbed in the i nh Figure of the fifth Experiment, and alfo
by the Light of the very bright Clouds next the Sun's
Body intermixing with thefe Colours, and by the Light
foartered by the inequalities in the pobfti of the Prifm, was
io very much compounded, that the Species which thofe
faint and dark Colours, the Indigo and Violet, call upon
the Paper were not diftindt enough to be well obferved.

Exper. o. A Prifm, wliofe two Angles at its Bafe were
equal to one another and half right ones, and the third
a right one, I placed in a beam, of the Sun's Light let in¬
to a dark Chamber through a hole in the Window 7 -flint
as in the thirAExperiment. And turning the Prifm flowly
about its Axis until all the Light which went through one
of its Angles and was refracted by it began to be reflected
by its Bale , at which till then it went out of the Glafs,

I obferved that thofe Rays which had fuffered the greateft
Refradtion were fooner refledted than the reft. I conceived
therefore that thofe Rays of the refledted Light, which
were moft Refrangible, did firft of all by a total Reflexion-
become more copious in that Light than the reft, and
that afterwards the reft alfo, by "a total Reflexion, be¬
came as copious as thefe. To try -this, I made the. re¬
flected Eight pafs through another Prifm, and being refra-

[ 38 ]

£ktd by it to fall afterwards upon a fheet of white Paper
placed at fome diftance behind it, and there by that Re¬
fraction to paint the ufual Colours of the Prifm. And
then caufing the firft Prifm to be turned about its Axis as
above, I obferved that when thofeRays which in this Prifm
had fuffered the greateft Refraction and appeared of a blew
and violet Colour began to be totally reflected, the blew
and violet Light on the Paper which was moft refraCted
in the fecond Prifm received a fenfible increafe above that
of the red and yellow, which was leaf! refraCted 5 and
afterwards when the reft of the Light which was green,
yellow and red began to be totally reflected in the firft
Prifm, the light of thofe Colours on the Paper received as
great an increafe as the violet and blew had done before.
Whence 'tis manifeft, that the beam of Light reflected by
the Bafe of the Prifm, being augmented firft by the more
Refrangible Rays and afterwards by the lefs Refrangible
ones, is compounded of Rays differently Refrangible.
And that all fuch reflected Light is of the fame Nature
with the Sun's Light, before its Incidence on the Bafe of
the Prifm, no Man ever doubted : it being generally al¬
lowed, that Light by fuch Reflexions fuffers no Alteration
in its Modifications and Properties. I do not here take
notice of any Refractions made in the Sides of the firft
Prifm, becaufe the Light enters it perpendicularly at the
firft Side, and goes out perpendicularly at the fecond Side,
and therefore fuffers none. So then, the Stiffs incident
Light being of the fame temper and conftitution with his
emergent Light, and the laft being compounded of Rays
differently Refrangible, the firft mull be in like manner
compounded.

Fig. 21. lllujlration. In the 21 dr Figure, A B C is the firft Prifm,
B C its Bafe, B and C its equal Angles at the Bafe, each

[ 39 ]

of 45 degrees, A its Rectangular Vertex, F M a beam of
the Sun’s Light let into a dark Room through a hole F
one third part of an Inch broad, M its Incidence on theBafe
of the Prifm, M G a lefs refra&ed Ray, M H a more refract¬
ed Ray, M N the beam of Light reflected from the Bafe,

V X Y the fecond Frifm by which this beam in palling
through it is refraCted, N t the lefs refracted Light of this
beam, and N p the more refracted part thereof. When the
firft Prifm A B C is turned about its Axis according to the
order of the Letters ABC, the Rays M H emerge more
and more obliquely out of that Prifm, and at length after
their moft oblique Emergence are reflected towards N,
and going on to p do increafe the number of the Rays N p*
Afterwards by continuing the motion of the firft Prifm, the
Rays MG are alfo reflected to N and increafe the number of
the Rays N t. And therefore the Light M N admits into
its Competition, firft the more Refrangible Rays, and then
the lefs Refrangible Rays, and yet after this Compofition
is of the lame; Nature with the Sun’s immediate Light F M, V
the Reflexion of the fpecular Bafe B C caufing no Altera¬
tion therein.

Exper. 10. Two Prifms, which were alike in Aiape, I
tied fo together, that their Axes and oppofite Sides being
Parallel, they compofed a Parallelepiped, And, the Sun
fliining into my dark Chamber through a little hole in the
Window-(hut, I placed that Parallelepiped in his beam at
fome diftance from the hole, in fuch a pofture that the Axes
of the Prifms might be perpendicular to the incident Rays,
and that thofe Rays being incident upon the firft Side of
one Priim, might go on through the two contiguous Sides
of both Prifms, and emerge out of the laft Side of the fe¬
cond Prifm. This Side being Parallel to the firft Side of
the firft Prifm, caufed the emerging Light to be Parallel

[ 4 ° ]

to the Incident. Then, beyond thefe two Prifms I placed
a third, which might refract that emergent Light, and by
that Refraction call: the ufnal Colours of the Prifm upon
the oppofite Wall, or upon a {heet of white Paper held at
a convenient diftance behind the Prifm for that refraCled
Light to fall upon it. After this I turned the Parallelepiped
about its Axis, and found that when the contiguous Sides
of the two Prifms beeame fo oblique to the incident Rays
that thofe Rays began all of them to be reflected, thofe
Rays which in the third Prifm had fuffered the greateft Re¬
fraction and painted the Paper with violet and blew, were
iirft of all by a total Reflexion taken out of the tranfmitted
Light, the reft remaining and on the Paper painting their
Colours of Green, Yellow, Orange, and Red as before 5
and afterwards by continuing the motion of the two Prifms,
the reft of the Rays alio by a total Reflexion vaniihed in
order, according to their degrees of Refrangibility. The
Light therefore which emerged out of the tw T o Prifms is
compounded of Rays differently Refrangible , feeing the
more Refrangible Rays may be taken out of it while the
lefs Refrangible remain. But this Light being trajeCted
only through the Parallel Superficies of the tw T o Prifms, if
it fuffered any change by the RefraCtion of one Superficies
it loft that impreffion by the contrary RefraCfcion of the
other Superficies, and fo being reftored to its priftine con-
ftitution became of the fame nature and condition as at firft
before its Incidence on thofe Prifms 5 and therefore, before
its Incidence, was as much compounded of Rays differently
Refrangible as afterwards.

lllujlration . In the 22th Figure ABC and B C D are the
the two Prifms tied together in the form of a Parallele¬
piped, their Sides BC and CB being contiguous, and
their Sides A B and C D Parallel. And H J K is the third

Prifm,

Priftn, by which the Suns Light propagated through the
hole F into the dark Chamber, and there paffing through
thofe fides of the Frifms AB, BC, CB and CD, is refra¬
cted at O to the white Paper PT, falling there partly upon
P by a greater Refraction, partly upon T by a lefs Refra¬
ction, and partly upon R and other intermediate places by
intermediate Refractions. By turning the Parallelopiped
ACBD about its Axis, according to the order of the Let¬
ters A,C,D,B, at length when the contiguous Planes BC
and CB become fufficiently oblique to the Rays F M,
which are incident upon them at M, there will vanifli to¬
tally out of the refracted Light OPT, firft of all the moft
refracted Rays O P, (the reft OR and O T remaining as
before) then the Rays O R and other intermediate ones,
and laftly, the leaft refracted Rays O T. For when the
Plane B C becomes fufficiently oblique to the Rays inci¬
dent upon it, thofe Rays will begin to be totally reflect¬
ed by it towards N 5 and firft the moft Refrangible Rays \\
will be totally reflected (as was explained in the preceding
experiment) and by confequence muft firft difappear at P,
and afterwards the reft as they are in order totally reflect¬
ed to N, they muft difappear in the fame order at R and
T. So then the Rays which at O fuffer the greateft Re¬
fraction, may be taken out of the Light M O whilft the reft
of the Rays remain in it, and therefore that Light MO is
Compounded of Rays differently Refrangible. And be-
caufe the Planes A B and C D are parallel, and therefore
by equal and contrary Refractions deftroy one anothers
Effects, the incident Light F M muft be of the fame kind
and nature with the emergent Light M O, and therefore
doth alfo confift of Rays differently Refrangible. Thefe
two Lights FM and MO,before the moft refrangible Rays
are feparated out of the emergent Light M O agree in Co-

F lour,

[ 4 2 ]

lour, and in all other properties fo far as my obfervation
reaches, and therefore are defervedly reputed of the fame
Nature and Conftitution, and by confequence the one is
compounded as well as the other. But after the molt: Re¬
frangible Rays begin to be totally reflected, and thereby
feparated out of the emergentLightMO,that Light changes
its Colour from white to a dilute and faint yellow, a pretty
good orange, a very full red fucceffively and then totally
vanifhes. For after the moil Refrangible Rays which paint
the Paper at P with a Purple Colour, are by a total re¬
flexion taken out of the Beam of light M O, the reft of
the Colours which appear on the Paper at R and T being
mixed in the light M O compound there a faint yellow,
and after the blue and part of the green which appear on
the Paper between P and R are taken away, the reft which
appear between R and T (that is the Yellow, Orange, Red
and a little Green) being mixed in the Beam M O com¬
pound there an Orange 5 and when all the Rays are by re¬
flexion taken out of the Beam MO, except the leaft Refran¬
gible, which at T appear of a full Red, their Colour is
the fame in that Beam M O as afterwards at T, the Re¬
fraction of the Prifm HJK ferving only to feparate the
differently Refrangible Rays, without making any alteration
in their Colours, as (hall be more fully proved hereafter.
All which confirms as well the firft Propofition as the fe-
cond.

Scholium . If this Experiment and the former be conjoyned
'fig. z 2 , and made one, by applying a fourth Prifm VXY to re¬
fract the reflected Beam MN towards tp, the conclufion
will be clearer. For then the light Np which in the 4th
Prifm is more refrafted, will become fuller and ftronger
when the Light O P, which in the third Prifm H J K is
more refracted, vanifhes at P 3 and afterwards when the lefs

refracted

refracted Light OT vaniflies at T, the lefs mfradedLight
N t will become encreafed whilft the more refraded Light
at p receives no further encreafe. And as the trajeded
Beam M O in vanifliing is always of fuch a Colour as
ought to refult from the mixture of the Colours which
fall upon the Paper P T. fo is the refleded Beam MN al¬
ways of fuch a Colour as ought to refult from the mix¬
ture of the Colours which fall upon the Paper pt. For
when the moil refrangible Rays are by a total Reflexion
taken out oi the Beam MO, and leave that Beam of an
Orange Colour, the excefs of thofe Rays in the refleded
Light, does not only make the Violet, Indigo and Blue at
p more full, but alfo makes the Beam M N change from
the yellowifli Colour of the Sun's Light, to a pale white in¬
clining to blue, and afterward recover its yellowifli Co¬
lour again, fo foon as all the reft of the tranfmitted light
MOT is refleded.

Now feeing that in all this variety of Experiments,
whether the trial be made inLight refleded,and that either
from natural Bodies, as in the firfl: and fecond Experiment,
or Specular, as in the Ninth 3 or in Light refraded, and
that either before the unequally refraded Rays are by di¬
verging feparated from one another, and lofing their white-
nefs which they have altogether, appear feverally of feve-
ral Colours, as in the fifth Experiment 3 or after they are
feparated from one another, and appear Coloured as in the
fixth,^ feventh, and eighth Experiments 3 or in Light tra-
jedea through Parallel fuperficies, deftroying each others
Effeds as in the 1 oth Experiment 3 there are always found
Rays, which at equal Incidences on the fame Medium fuf-
fer unequal Refradions, and that without any Iplitting or
dilating of fingle Rays, or contingence in the inequality
or the Refradions, as is proved in the fifth and fixth Ex-

F 2 periments 3

[ 44 ]

periments $ and feeing the Rays which differ in Refrangibi-
lity may be parted and forted from one another, and that
either by Refraction as in the third Experiment, or by Re¬
flexion as in the tenth, and then the feveral forts apart at
equal Incidences fuffer unequal Refractions, and thofe forts
are more refraCted than others after feparation, which were
more refraCted before it, as in the fixth and following Ex¬
periments, and if the Suns Light be trajeCted through three
or more crofs Prifms fucceffively, thofe Rays which in the
firft Prifm are refraCted more than others are in all the fol¬
lowing Prifms, refraCted more then others in the fame rate
and proportion, as appears by the fifth Experiment 3 it’s
manifeft that the Sun’s Light is an Heterogeneous mixture of
Rays, fome of which are conftantly more Refrangible then
others, as was to be propofed.

PROP. III. Theor. III.

'The Suns Light conflfts of flays differing in fleflexibility, and
thofe (flays are more fleflexible than others which are more
frangible .

T HIS is manifeft by the ninth and tenth Experi¬
ments : For in the ninth Experiment, by turning
the Prifm about its Axis, until the Rays within it which in
going out into the Air were refraCted by its Bafe, became
fo oblique to that Bafe, as to begin to be totally reflected
thereby 5 thofe Rays became firft of all totally reflected,
which before at equal Incidences with the reft had fuffered
the greateft RefraCtion. And the fame thing happens in
the Reflexion made by the commonBafe of the two Prifms
in the tenth Experiment.

<P fl^Of.

[ 45 1

PROP. IV. Prob. I.

To feparate from one another the Heterogeneous 'ays of

Compound Light .

np H E Heterogeneous Rays are in fome meafure fepa-
j| rated from one another by the Refra&ion of the
Prifm in the third Experiment, and in the fifth Experiment
by taking away the Penumbra from the Rectilinear fides of
the Coloured Image, that feparation in thofe very Rectili¬
near fides or ftraight edges of the Image becomes perfect.

But in all places between thofe rectilinear edges, thofe in¬
numerable Circles there defcribed, which are lever ally illu¬
minated by Homogeneral Rays, by interfering with one
another, and being every where commixt, do render the
Light fufficiently Compound. But if thefe Circles, whilft
their Centers keep their diftances and pofitions, could be \
made lefs in Diameter, their interfering one with another
and by confequence the mixture of the Heterogeneous
Rays would be proportionally diminifhed. In the 23th Fig. 23.
Figure let A^G, B H, C J, D K, EL, F M be the Circles
which fo many forts of Rays flowing from the fame Difque
of the Sun, do in the third Experiment illuminate 5 of all
which and innumerable other intermediate ones lying in a
continual Series between the two Rectilinear and Parallel
edg es of the Sun s oblong Image P T, that Image is com-
pofed as was explained in the fifth Experiment. And let
ag, hh y ci , dkj el , fm be fo many lefs Circles lying in
a like continual Series between two Parallel right Lines af
and g m with the fame diftances between their Centers,
and illuminated by the fame forts of Rays, that is the
Circle ag with the fame fort by which the correfponding

Circle

[4<n

Circle AG was illuminated, and the Circle bh with the fame
fort by which the correfponding Circle BHwas illuminated,
and the reft of the Circles c z, d 4 , el, fm refpedtively,
with the fame forts of Rays by which the feveral corre¬
fponding Circles C J, D K, EL, FM were illuminated.
In the Figure P T compofed of the greater Circles, three
of thofe Circles AG, B H, CJ, are fo expanded into one
another, that the three forts of Rays by which thofe Cir¬
cles are illuminated, together with other innumerable forts
of intermediate Rays, are mixed at Q.R in the middle of
the Circle B H. And the like mixture happens through¬
out almoft the whole length of the Figure P T. But in
the Figure p t compofed of the lefs Circles, the three lefs
Circles ag, bb, c i , which anfwer to thofe three greater, do
not extend into one another 3 nor are there any where
mingled io much as any two of the three forts of Rays
by which thofe Circles are illuminated, and which in the
Figure P T are all of them intermingled at B H.

Now he that fhall thus confider it, will eafily underftand
that the mixture is diminifhed in the fame Proportion
with the Diameters of the Circles. If the Diameters of
the Circles whilft their Centers remain the fame, be made
three times lefs than before, the mixture will be alfo three
times lefs 5 if ten times lefs, the mixture will be ten times
lefs, and fo of other Proportions. That is, the mixture
of the Rays in the greater Figure P T will be to their mix¬
ture in the lefs p t , as the Latitude of the greater Figure is
to the Latitude of the lefs. For the Latitudes of thefe Fi¬
gures are equal to the Diameters of their Circles. And
hence it eafily follows, that the mixture of the Rays in the
refradted Spedtrum p t is to the mixture of the Rays in the
diredt and immediate Light of the Sun, as the breadth of
that Spedtrum is to .he difference between the length and
breadth of the fame Spedtrum. So

So then, if we would diminifli the mhtmt of the Rays',
we are to diminifli the Diameters of the Circles. Now
thefe would be diminiflied if the Sun's Diameter to which
they anfwer could be made lefs than it-is, or (which comes
to the fame purpofe) if without Doors, at a great diftance
from the Prifm towards the Sun, feme opake body were
placed, with a round hole in the middle of it, to intercept
all the Sun's Light, excepting fo much as coming from
the middle of his Body could pafs through that hole to
the Prifm. For fo the Circles AG, B H and the reft,
would not any longer anfwer to the whole Difque of the
Sun , but only to that part of it which could be feen
from the Prifm through that hole, that is to the apparent
magnitude of that hole viewed from the Prifm. But that
thefe Circles may anfwer more diftindly to that hole a
Lens is to be placed by the Prifm to caft the Image of the
hole, (that is, every one of the Circles A G, B H, <&c.) di¬
ftindly upon the Paper at P T, after fuch a manner as by
a Lens placed at a Window the Species of Objeds abroad
are caft diftindly upon a Paper within the Room, and the
Redilinear Sides of the oblong folar Image in the fifth
Experiment became diftind without any Penumbra. If
this be done it will not be neceffary to place that hole
very far off, no not beyond the Window. And therefore
inftead of that hole, I ufed the hole in the Window-fliut
as follows.

Exper. it. In the Sun's Light let into my darkned
Chamber through a fmall round hole in my Window-
fliut, at about i o or 12 Feet from the Window, I placed
a Lens, by which the Image of the hole might be di¬
ftindiy caft upon a fheet of white Paper, placed at the
diftance of fix, eight, ten or twelve Feet from the Lens,
For according to the difference of the Lenfes I ufed various

diftances,

[+ 8 ]

diftances, which I think not worth the while to defcribe.
Then immediately after the Lens I placed a Prifm, by
which the trajeded Light might be refraded either up¬
wards or tideways, and thereby the round Image which
the Lens alone did call upon the Paper might be drawn
out into a long one with Parallel Sides , as in the third
Experiment. This oblong Image I let fall upon another
Paper at about the fame diftance from the Prifm as be¬
fore, moving the Paper either towards the Prifm or from
it, until I found the juft diftance where the Rectilinear
Sides of the Image became mod diftind. For in this cafe
the circular Images of the hole which compofe that Image
after the fame manner that the Circles ag J bh , ci, Sec. do
x 3. the Figure p t , were terminated mod diftindly without any
Penumbra, and therefore extended into one another the
leaft that they could, and by confequence the mixture of
the Heterogeneous Rays was now the leaft of all. By this
2 3, means I ufed to form an oblong Image (fuch as is p t ) of
24. circularlmages of the hole (fuch as are ag, bh , ci, See.)
and by ufing a greater or lefs hole in the Window-fhut, I
made the circular Images ag, b /;, c /, See. of which it was
formed, to become greater or lefs at pleafure, and thereby
the mixture of the Rays in the Image p t to be as much
or as little as I defired.

24. Illufiration . In the 24th Figure, F reprefents the circular
hole in the Window^Ihut, M N the Lens whereby the
Image or Species of that hole is caft diftindly upon a
Paper at J, ABC the Prifm whereby the Rays are at their
emerging out of the Lens refraded from J towards ano¬
ther Paper at p t , and the round Image at J is turned into
an oblong Image p t falling on that other Paper. This
Image p t confifts of Circles placed one after another in a
Redilinear order, as was fufficiently explained in the fifth

Experiment 3

[ 49 ]

Experiment • and thefe Circles are equal to the Circle X,
and confequently anfwer in Magnitude to the hole F 5 and
therefore by diminishing that hole they may be at pleafure
diminished, whifft their Centers remain in their places.
By this means I made the breadth of the Image p t to be
forty times, and fometimes fixty or Seventy times lefs than
its length. As for inftance, if the breadth of the hole f
be of an Inch, and MF the diftance of the Lens from
the hole be 1 2 Feet 5 and if p B or p M the diftance of
the Image pt from the Prifm or Lens be 10 Feet, and the
refra&ing Angle of the Prifm b t 62 degrees, the breadth
of the Image p t will be ~ of an Inch and the length about
fix Inches, and therefore the length to the breadth as 72
to 1, and by confequence the Light of this Image 71 times
lefs compound than the Sun’s dired: Light. And Light
thus far Simple and Homogeneal, is fumcient for trying
all the Experiments in this Book about fimple Light. For
the compofition of Heterogeneal Rays is in this Light fo
little that it is fcarce to be difcovered and perceived by
fenfe, except perhaps in the Indigo and Violet 5 for thefe
being dark Colours, do eafily fuffer a fenfible allay by that
little fcattering Light which ufes to be refrafted irregularly
by the inequaliteis of the Prifm.

Yet inftead of the circular hole F, tis better to fubfti-
tute an oblong hole Shaped like a long Parallelogram
with its length Parallel to the Prifm ABC. For if this
hole be an Inch or two long, and but a tenth or twentieth
part of an Inch broad or narrower : the Light of the Image
p t will be as Simple as before or Simpler, and the Image
will become much broader, and therefore more fit to have
Experiments tried in its Light than before.

Inftead of this Parallelogram-hole may be fubftitured a
Triangular one or equal Sides, whofe Bafe for inftance is

G about

!>j

about the tenth part of an Inch, and its height an Inch or
more. For by this means, if the Axis of the Prifm be
Parallel to the Perpendicular of the Triangle , the Image
5. pt will now be formed of Equicrural Triangles ag, bh y ci ,
e 1^ f m , &cc. and innumerable other intermediate ones
anfwering to the Triangular hole in flhape and bignefs,and
lying one after another in a continual Series between two
Parallel Lines af an Agm. Thefe Triangles are a little
intermingled at their Bafes but not at their Vertices, and
therefore the Light on the brighter fide af of the Image
where the Bafes of the Triangles are is a little compounded,
but on the darker fide^ m is altogether uncompounded,
and in all places between the fides the Compofition is
Proportional to the diftances of the places from that ob-
fcurer fide g m. And having a Spedtrum pt of fuch a
Compofition, we may try Experiments either in its ftronger
and lefs fimple Light near the fide af or in its weaker
and Ampler Light near the other fide l m , as it fihall feem
moft convenient.

But in making Experiments of this kind the Chamber
ought to be made as dark as can be, leaft any forreign
Light mingle it felf with the Light of the Spectrum p
and render it compound 3 efpecially if we would try Ex¬
periments in the more fimple Light next the fide g m of
the Spedtrun^ which being fainter, will have a lefs Pro¬
portion to the forreign Light, and fo by the mixture of
that Light be more troubled and made more compound.
The Lens alfo ought to be good, fuch as may ferve for
Optical Ufes, and the Prifm ought to have a large Angle,
fuppofe of 70 degrees, and to be well wrought, being
made of Glafs free from Bubbles and Veins, with its fides
not a little Convex or Concave as ufually happens but
truly Plane,and its pollifh elaborate, as in working Optick-

c$o

glaffes , and not fuch as is ufually wrought with Putty,
whereby the edges of the Sand-holes being worn away,
there are left all over the Glafs a numberlefs company of
very little Convex polite rifings like Waves. The edges
alfo of the Prifrn and Lens fo far as they may make any
irregular Refradion, muft be covered with a black Paper
glewed on. And all the Light of the Suns beam let into
the Chamber which is ufelefs and unprofitable to the Ex¬
periment, ought to be intercepted with black Paper or other
black Obftacles. For otherwife the ufelefs Light being
refleded every way in the Chamber, will mix with the
oblong Spedrum and help to difturb it. In trying thefe
things fo much Diligence is not altogether neceffary, but
it will promote the fuccefs of the Experiments, and by a
very fcrupulous Examiner of things deferves to be applied*
Its difficult to get glafs Prifms fit for this purpofe, and
and therefore I ufed fometimes Prifmatick Venels made
with pieces of broken Looking-glades, and filled with rain
Water. And to increafe the Refradion, I fometimes im¬
pregnated the Water ftrongly with Saccharum SaturuL

PROP . V. Theor. IV.

Homogeneal Light is refralied regularly without any Dilatation
fplitting or /battering 0/ the [ays , and the confu/ed Vifion
of Objects feen through <%efratting Ladies by Heterogeneal
Light ari/es from the different Lgfrangibihty of federal forts
of (ffays.

T H E firft Part of this Propofition has been already
diffidently proved in the fifth Experiment, and will
further appear by the Experiments which follow.

G 2

Exper . t2.

Exper . i 2a In the middle of a black Paper I made a
round hole about a fifth or fixth part of an Inch in Dia¬
meter. Upon this Paper I caufed the Spectrum of Homo-
geneal Light defcribed in the former Propofition, fo to
fall, that forne part of the Light might pafs through the
hole of the Paper. This tranlmitted part of the Light I
refraded with a Prifm placed behind the Paper, and let¬
ting this refra£fed Light fall perpendicularly upon a white
Paper two or three Feet diftant from the Priftn, I found
that the Spedtrum formed on the Paper by this Light was
not oblong, as when 'tis made (in the third Experiment)
by Refrafting the Suns compound Light, but was (fo far
as I could judge by my Eye) perfectly circular, the length
being no greater than the breadth. Which fhews that this
Light is refradted regularly without any Dilatation of the
Rays.

Exper . 15. In the Homogeneal Light I placed a Circle
of ^ of an Inch in Diameter, and in the Sun’s unrefradted
Heterogeneal white Light I placed another Paper Circle of
the fame bignefs. And going from the Papers to the diftance
offomeFeet, I viewed both Circles through a Prifm. The
Circle illuminated by the Sun's Heterogeneal Light appear¬
ed very oblong as in the fourth Experiment, the length
being many times greater than the breadth : but the other
Circle illuminated with Homogeneal Light appeared Cir¬
cular and diftindtly defined as when 'tis viewed with the
naked Eye. Which proves the whole Propofition.

Exper . 14. In the Homogeneal Light I placed Flies and
fuch like Minute Objedls, and viewing them through a
Prifm, I faw their Parts as diftindtly defined as if I had
viewed them with the naked Eye. The fame Objects pla¬
ced in the Sun’s unrefradted Heterogeneal Light which was
white I viewed alfo through a Prifm, and faw them moil

confufedly

eonfufedly defined,fo that I could not diftinguifii their fmal-
ler Parts from one another. I placed alfo the Letters of a
fmall Print one while in the Homogeneal Light and then
in the Heterogeneal, and viewing them through a Prifm,
they appeared in the latter cafe fo confufed and indiftinCt
that I could not read them 3 but in the former they ap¬
peared fo diftinCt that I could read readily, and thought
I faw them as diftinCt as when I viewed them with my
naked Eye. In both cafes I viewed the fame Objects
through the fame Prifm at the fame diftance from me and
in the fame Situation. There was no difference but in the
Light by which the Objects were illuminated, and which
in one cafe was Simple and in the other Compound, and
therefore the diftinCt Vifion in the former cafe and confu¬
fed in the latter could arife from nothing elfe than from
that difference of the Lights. Which proves the whole
Propofition.

And in thefe three Experiments it is further very remar¬
kable, that the Colour of Homogeneal Light was never
changed by the Refraction.

PROP. VI. Theor. V.

The Sine of Incidence of every [ay confidered apart , is to its Sim ’

of fRefraBwn in a given I^atio*

T HAT every Ray confidered apart is conftant to
it felf in fame certain degree of Refrangibility, is
fufficiently manifeft out of what has been faid. Thofe
Rays which in the firft RefraCtion are at equal Incidences
mo ft refraCted, are alfo in the following Refractions at
equal Incidences mod refraCted 3 and fo of the leaft Re¬
frangible , and the reft which have any mean degree of

Refran-

Refrangibility, as is tnanifeft by the 5 th, 6th, 7th, 8th,
and 9th Experiments. And thofe which the firft time at
like Incidences are equally refracted, are again at like In¬
cidences equally and uniformly refra&ed, and that whe¬
ther they be refraCted before they be feparated from one
another as in the 5 th Experiment, or whether they be re¬
fracted apart, as in the 1 2 th, 13 th and 14th Experiments.
The RefraCtion therefore of every Ray apart is regular,
and what Rule that RefraCtion obferves we are now
to fliew.

The late Writers in Opticks teach, that the Sines of In¬
cidence are in a given Proportion to the Sines of Refra¬
ction, as was explained in the 5 th Axiom 3 and fome by
Inftruments fitted for meafuring Refractions, or otherwise
experimentally examining this Proportion, do acquaint us
that they have found it accurate. But whilft they, not
underftanding the different Refrangibility of feveral Rays,
conceived them all to be refraCted according to one and
the fame Proportion, 5 tis to be prefumed that they adapted
their Meafures only to the middle of the refraCted Light 3
fo that from their Meafures we may conclude only that
the Rays which have a mean degree of Refrangibility ,
that is thofe which when feparated from the reft appear
green, are refraCted according to a given Proportion of
their Sines. And therefore we are now to fliew that the
like given Proportions obtain in all the reft. That it
fhould be fo is very reafonable, Nature being ever confor¬
mable to her felf: but an experimental Proof is defired.
And fuch a Proof will be had if we can fliew that the
Sines of RefraCtion of Rays differently Refrangible are
one to another in a given Proportion when their Sines of
Incidence are equal. For if the Sines of RefraCtion of all
the Rays are in given Proportions to the Sine of RefraCtion

LS5]

of a Ray which has a mean degree of Reffangibilky, and
this Sine is in a given Proportion to the equal Sines of
Incidence, thofe other Sines of Refraction will alfo be in
given Proportions to the equal Sines of Incidence. Now
when the Sines of Incidence are equal, it will appear by
the following Experiment that the Sines of Refradion are
in a given Proportion to one another.

Exper. 15. The Sun Alining into a dark Chamber
through a little round hole in the Window-fhut, let S re-Tg*.
prefent his round white Image painted on the oppofite
Wall by his dired Light, P T his oblong coloured Image
made by refrading that Light with a Prifm placed at the
Window 3 and pt, or ip it, or 3/? 3 t y his oblong coloured
Image made by refrading again the fame Light tideways
with a fecond Prifm placed immediately after the firft in
a crofs Pofition to it, as was explained in the fifth Experi¬
ment : that is to fay, pt when the Refradion of the fecond
Prifm is fmall, 2 p 21 when its Refradion is greater, and
3 pit when it is greateft. For fuch will be the diverfity
of the Kefradions if the refrading Angle of the fecond
Prifm be of various Magnitudes 3 fuppofe of fifteen or
twenty degrees to make the Image p t , of thirty or
forty to make the Image ip zt, and of fixty to make,
the Image 3 p 3 1. But for want of folid Glafs Prifms with
Angles of convenient bigneffes, there may be Veffels
made of poliflied Plates of Glafs cemented together in the
form of Prifms and filled with Water. Thefe things being
thus ordered, I observed that; all the folar images or co¬
loured Spedrums PT, pt, 2p 2t, %p did very nearly
converge to the place S on which the dired Light of the
Sun fell and painted his white round Image when the
Prifms were .taken away. The Axis of the Spedrum PT,
that is the Line, drawn through the middle of it Parallel to

C5«]

its Redilinear Sides, did when produced pafs exactly through
the middle of that white round Image S. And when the
Refraction of the fecond Prifm was equal to the Refraction
of the firll, the refracting Angles of them both being about
do degrees, the Axis of the Spectrum ip 3 £ made by that
Refraction, did when produced p^fs alfo through the mid¬
dle of the fame white round Image S. But when the Re¬
fraction of the fecond Prifm was lefs than that of the firft,
the produced Axes of the SpeCtrums tp or it zp made
by that RefraCtion did cut the produced Axis of the Spe¬
ctrum TP in the Points m and ft, a little beyond the Cen¬
ter of that white round Image S. Whence the Proportion
of the Line ^Tto the Line 3^ P was a little greater than
the Proportion of z t T to 2 pP } and this Proportion a little
greater than that of fT topP. Now when the Light of
the SpeCtrum P T falls perpendicularly upon the Wall, thofe
Lines 3 1 T, 3 p P, and 2 1 T, ip P and t T, p P,are the Tan-
gents of the Refractions 3 and therefore by this Experiment
the Proportions of the Tangents of the Refractions are ob¬
tained, from whence the Proportions of the Sines being deriv¬
ed, they come out equal, fo far as by viewing the SpeCtrums
and ufing fome Mathematical reafoning I could Eftimate.
Eor I did not make an Accurate Computation. So then
the Propofition holds true in every Ray apart, fo far as ap¬
pears by Experiment. And that it is accurately true may
be demonftrated upon this Suppofition, That Bodies ref rah
Light hy acting upon its (fays in Lines Perpendicular to their
Surfaces . But in order to this Demonftration, I mull di-
ftinguilh the Motion of every Ray into two Motions, the
one Perpendicular to the refracting Surface, the other Pa¬
rallel to it, and concerning the Perpendicular Motion lay
down the following Propofition.

If any Motion or moving thing whatfoever be incident
with any velocity on any broad and thin Space termina¬
ted on both fides by two Parallel Planes, and in its paflage
through that fpace be urged perpendicularly towards the
further Plane by any force which at given diftances from
the Plane is of given quantities 5 the perpendicular Velo¬
city of that Motion or Thing, at its emerging out of that
fpace, fhall be always equal to the Square Root of the
Summ of the Square of the perpendicular Velocity of
that Motion or Thing at its Incidence on that fpace $
and of the Square of the perpendicular Velocity which
that Motion or Thing would have at its Emergence, if
at its Incidence its perpendicular Velocity was infinitely
little.

And the fame Propofition holds true of any*Motion or
Thing perpendicularly retarded in its paflage through that
(pace, if inftead of the Summ of the two Squares you take
their difference. The Demonftration Mathematicians will
eafily find out, and therefore I fhall not trouble the Rea¬
der with it,

Suppofe now that a Ray coming moll obliquely in the Fig.
Line MC be refra&ed at C by the Plane RS into the Line
CN, and if it be required to find the Line CE into which
any other Ray AC fhall be refracted 3 let MC, AD, be
the Sines of incidence of the two Rays, and NG, EF, their
Sines of Refraction, and let the equal Motions of the In¬
cident Rays be reprefented by the equal Lines M C and
AC, and the Motion MC being confidered as parallel to
the refracting Plane, let the other Motion AC be diftin-
guifihed into two Motions. AD; and DC, one of winch
AD is parallel, and she other DC perpendicular to the re¬
fracting Surface, hi like manner, let the Motions of the
emering Rays be diftlngniflfd into two, whereof the per-

H pendicular

perpendicular ones are ^ CG and ^ CF. And if the

force of the refracting Plane begins to aCt upon the Rays
either in that Plane or at a certain diftance from it on the
one fide, and ends at a certain diftance from it on the
other fide, and in all places between thofe two Limits aCts
upon the Rays in Lines perpendicular to that rafrafting
Plane, and the Actions upon the Rays at equal diftances
from the refracting Plane be equal, and at unequal ones ei¬
ther equal or unequal according to any rate whatever •
that motion of the Ray which is Parallel to the refracting
Plane will fuffer no alteration by that force 5 and that mo¬
tion which is perpendicular to it will be altered according
to the rule of the foregoing Propofition. If therefore for
the perpendicular Velocity of the emerging Ray CN you

write ^ CG as above, then the perpendicular Velocity
of any other emerging Ray CE which was ^ CF, will be

equal to the fquare Root of CDq + CGq. And

by fquaring thefe equals, and adding to them the Equals
ADq and MCq —CDand dividing the Summs by the
Equals CF$ + EF q and CG^-|-NGf, you will have

equal to Whence AD, the Sine of Incidence,

is to EF the Sine of RefraCtion, as MC to NG, that is,
in a given ratio . And this Demonfiration being general,
without determining what Light is, or by what kind of
force it is refraCted, or affuming any thing further than
that the refracting Body aCts upon the Rays in Lines per¬
pendicular to its Surface 3 I take it to be a very convincing
Argument of the full Truth of this Propofition*

So then* if the ratio of the Sines of Incidence and Re-
fra&ion of any fort of Rays be found in any one Cafe, us
given in all Cafes 3 and this may be readily found by the
Method in the following Propofition.

PROP. VII Theor. VI

‘The (Perfe&ion ofTelefcopes is impeded by the different %efran~

gibility of the Ofays of Light,

rpH E imperfection of Telefcopes is vulgarly attri-
I buted to the fpherical Figures of the Glaffes, and
therefore Mathematicians have propounded to Figure them
by the Conical Sections. To Chew that they are mifta-
ken, I have inferred this Propofition^ the truth of which
will appear by the meafures of the Refractions of the feve-
ral forts of Rays 5 and thefe meafures I thus determine*

In the third experiment of the firft Book, where the re¬
fracting Angle of the Prifrn was 6 z] degrees, the half of

that Angle 31 deg. 1 5 min. is the Angle of Incidence of
the Rays at their going out of the Glafs into the Air 3 and
the Sine of this Angle is 5188, the Radius being ioooo.
When the Axis of this Prifrn was parallel to the Horizon,
and the Refraction of the Rays at their Incidence on this
Prifrn equal to that at their Emergence out of it, I obferved
with a Qiiadrant the Angle which the mean refrangible Rays
(that is, thofe which went to the middle ofthe Sun's colour¬
ed Image ) made with the Horizon and by this Angle and
th^ Sun s altitude oblerved at the fame time, I found the
Angle which the emergent Rays contained with the incident
to be 44 deg. and 40 min. and the half of this Aiwle ^d-
ded to the Angle of Incidence 3 1 deg. 15 min. males the

H t Antrim

[So]'

Angle of RefraCtion,which is therefore 5 3 degi 3 j min. and
its Sine 8047. Thefe are the Sines of Incidence and Re¬
fraCtion of the mean refrangible Rays, and their proportion
in round numbers is 20 to 31. This Glafs was of a colouring
dining to green. The laft of the Prifms mentioned in the
third Experiment was of clear white Glafs. Its reloading
Angle 63 i degrees. The Angle which the emergent Rays
contained, with the incident 45 deg. 50 min. The Sine of
half the firft Angle 5 2dr. The Sine of half the Summ
of the Angles 815 7. And their proportion in round num¬
bers 20 to 31 as before.

From the Length of the Image, which was about 9-! or

1 o Inches, fubduCt its Breadth, which was 2 ~ Inches, and
the Remainder 7* Inches would be the length of the Image
were the Sun but a point, and therefore fubtends the An¬
gle which the moft and lead: refrangible Rays, when inci¬
dent on the Prifm in the fame Lines, do contain with one
another after their Emergence. Whence this Angle is

2 deg. 0/ 7." For the diftance between the Image and the
Prifm where this Angle is made, was 18| Feet, and at that
diftance the Chord /j Inches fubtends an Angle of 2 deg.
0/ 7." Now half this Angle is the Angle which thefe e-
mergent Rays contain with the emergent mean refrangible
Rays, and a quarter thereof, that is 30. 2," may be ac¬
counted the Angle which they would contain which the
fame emergent mean refrangible Rays, were they co-inci¬
dent to them within the Glafs and differed no other Re-
fraCtiomthen that at their Emergence. For if two equal
Refractions, the one at the incidence of the Rays on the
Prifm, the other at their Emergence, make half the Angle
2 deg. of 7." then one of thofe Refractions will make
about a quarter of that Angle, and this quarter added to

and '

[S']

and fubduded from the Angle of Refradion of the mean,
refrangible Rays, which was 53 deg* 35", gives the An¬
gles of Refradion of the moft and leaft refrangible Rays
54 deg. 5* z", and 53 deg. 4 5 8 ^ whofe Sines are 8099
and 7995, the common Angle of Incidence being 3 1 deg.

1 5' and its Sine 51885 and thefe Sines in the leaft round
numbers are in proportion to one another as 78 and 77
to 50.

Now if you fubdud the common Sine of Incidence 501
from the Sines of Refradion 77 and 78, the remainders
27 and 28 fhew that in fmall Refradions the Refradion >
of the leaft refrangible Rays is to the Refradion of the moft
refrangible ones as 27 to 28 very nearly, and that the dif¬
ference of the Refractions of the leaft refrangible and moft .
refrangible Rays is about the 27{th part of the whole Re-*

fraction of the mean refrangible Rays.

Whence they that are skilled in Opcicks will eafily un-
derftand, that the breadth of the leaft circular fpace into
which Object' Glades of Telefcopes can collect all forts of
Parallel Rays, is about the 27-th part of half the aperture
of the Glafs, or 5 5 th part of the whole aperture 5 and
that the Focus of the moft refrangible Rays is nearer to the
Object-Glafs than the Focus of the leaft refrangible ones, by
about the 27jth part of the diftance between the Object-
Glafs and the Focus of the mean refrangible ones.

And if Rays of all forts,flowing from any one lucid point
in the Axis of any convex Lens, be made by the Refraction ^
of the Lens to converge to points not too remote from the
Lens, the Focus of the moft refrangible Rays flhall be
nearer to the Lens than the Focus of the leaft refrangible
ones, by a diftance which is to the 27^th part of the di¬
ftance of the Focus of the mean refrangible Rays from the
Lens as the diftance between that Focus and the lucid

point....

■<; 1

point from whence the Rays flow is to die diftance be¬
tween that lucid point and the Lens very nearly.

Now to examine whether the difference between the Re-
fractions which the moft refrangible and the leaft refran¬
gible Rays flowing from the fame point fuffer in the Ob-
jed-Glalies of Telefcopes and fuch like Glafles, be fo great
, as is here defcribed, I contrived the following Experi-
: ment.

Exper. 1 6. The Lens which I ufed in the fecond and
eighth Experiments, being placed fix Feet and an Inch dif-
tant from any Objed, colleded the Species of that Object
by the mean refrangible Rays at the diftance of fix Feet
and an Inch from the Lens on the other fide. And there¬
fore by the foregoing Rule it ought to colled the Species of
that Objed by the leaft refrangible Rays at the diftance of
Fix Feet and 31 Inches from the Lens, and by the moft re¬
frangible ones at the diftance of five Feet and lof Inches
from it: So that between the tw o Places where thefe leaft
and moft refrangible Rays colled the Species, there may
be the diftance of about 5* Inches. For by that Rule, as
tfix Feet and an Inch ( the diftance of the Lens from the
lucid Object) is to twelve Feet and two Inches (the di¬
ftance of the lucid Object from the Focus of the mean re¬
frangible Rays) that is, as one is to two, fo is the 17 [th
part of fix Feet and an Inch (the diftance between the Lens
.and the fameFocus ) to the diftance between the Focus of
the moft refrangible Rays and the Focus of the leaft re¬
frangible ones, which is therefore 5- inches, that is very

nearly 5 ~ Inches. Now to know whether this meafure
was true, I repeated the fecond and eighth Experiment of
this Book with coloured Light, which was lefs compound¬
ed than that I there made me of: For I now feparated the

hetero-

fi eterogencous Rays from one another by the Method I de~
fc ribed in the i itn Experiment, fo as to make a coloured
Spedrum about twelve or fifteen times longer than broach
This Spedrum I call on a printed book, and placing the
above-mentioned Lens at the diftance of fix Feet and an
Inch from this Spedrum to colled the Species of the illu¬
minated Letters at the fame diftance on the other fide, I
found that the Species of the Letters illuminated with Blue
were nearer to the Lens than thofe illuminated with deep
Red by about three Inches or three and a quarter: but the
Species of the Letters illuminated with Indigo and Violet
appeared fo confufed and indiftind, that I could not read
them : Whereupon viewing the Prifm, I found it was full
of Veins running from one end of the Glafs to the other 5
fo that the Refradion could not be regular. I took ano¬
ther Prifm therefore which was free from Veins, and in-
ftead of the Letters I ufed two or three Parallel black Lines
a little broader than the ftroakes of the Letters, and call¬
ing the Colours upon thefe Lines in fuch manner that the
Lines ran along die Colours from one end of the Spedum
to the other, I found that the Focus where the Indigo, or
confine of this colour and Violet call the Species of the
black Lines mort diftindly,tobe about 4 Inches or 4* near¬
er to the Lens than the Focus where the deepeft Red caft
the Species of the fame black Lines moft diftindfy.
The violet was fo faint and dark, that I could not
difeern the Species of the Lines diPdnctly by that Co¬
lour 5 and therefore confidering that the Prifm was made
©f a dark coloured Glafs inclining to Green, I took another
Pifm of clear white Glafs 3 but the Spedrum of Colours
which this Prifm made had long white Streams of faint
Light Shooting out from both ends of the Colours, which
made me conclude that fomediing was axmifs 3 and view-

t«4]

ing the Prifm, 1 found two or three little Bubbles in the
Glafs which refracted the Light irregularly. Wherefore I
covered that part of the Glafs 5 with black Paper, and let¬
ting the Light pais through another part of it which was
free from fiich Bubles, the Spectrum of Colours became
free from thofe irregular Streams of Light, and was now
fuch as I defired. But Mill I found the Violet fo dark and
faint, that I could fcarcefee the Species of the Lines by the
Violet, and not at all by the deepeft part of it, which was
next the end of the Spedlrum. I fulpedted therefore that
this faint and dark Colour might be allayed by that Mat¬
tering Light which was refracted, and refledted irregularly
partly by fome very final! Bubbles in the Glaffes and
partly by the inequalities of their Poliffr. which Light,
tho 5 it was but little, yet it being of a White Colour,
might fuffice to affedt the Senfe fo ftrongly as to difturb
the Phenomena of that weak and dark Colour the Violet,
and therefore I tried, as in the 12th, 13 th, 14th Experi¬
ments, whether the Light of this Colour did not confift of
a fenfible mixture of heterogeneous Rays, but found it did
not. Nor did the Refradtions caufe any other fenfible
Colour than Violet to emerge out of this Light, as they
would have done out of White Light, and by con-
fequence out of this Violet Light had it been fenfi-
bly compounded with White Light. And therefore I con¬
cluded, that the reafon why I could not fee the Species of
the Lines diftindtly by this Colour, was only the darknefs
of this Colour and Thinnefs of its Light, and its dif-
tance from the Axis of the Lens 3 I divided therefore thofe
Parallel Black Lines into equal Parts, by which I might
readily know the diltances of the Colours in the Spedfrum
from one another, and noted the diftances of the Lens
from the Foci of fuch Colours as caft the Species of the

Lines

Lilies diflin&ly, and then confidered whether the diffe¬
rence of thofe diflances bear fuch proportion to 5 ^Inches,
the greatefl difference of the diflances which the Foci of
the deepefl Red and Violet ought to have from the Lens,
as the diflance of the obferved Colours from one another
in the Spedlrum bear to the like diflance of the deepefl Red
and Violet meafured in the rectilinear fides of the Spect¬
rum, that is, to the length of thofe fides or excefs of the
length of the Spectrum above its breadth. And my Ob-
fervations were as follows.

When I obferved and compared the deepefl fenfibleRed,
and the Colour in the confine of Green and Blue, which
at that rectilinear fides of the Spectrum was diflant from it
half the length of thofe fides, the Focus where the confine
of Green and Blue call the Species of the Lines diflinCtly
on the Paper, was nearer to the Lens then the Focus where
the Red cafl thofe Lines diCtinCtly on it by about or

2 - Inches. For fometimes the Meafures were a little grea¬
ter, fometimes a little lefs, but feldom varied from one
another above ~ of an Inch. For it was very difficult to
define the Places of the Foci, without fome little Errors.
Now if the Colours diflant half the length of the Image,

{ meafured at its rectilinear fides) give z~ or 2 ~ difference
of the diflances of their Foci from the Lens, then the Co¬
lours diflant the whole length ought to give j or 5^ Inches
difference of thofe diflances.

But here its to be noted, that I could not fee the Red
to the full End of the SpeCtrum, but only to the Center
of the Semicircle which bounded that End, or a little far¬
ther 5 and therefore I compared this Red not with that Co¬
lour which was exadly in the middle of the Spectrum, or
confine of Green and Blue, but with that which verged a
little more to the Blue than to the Green : And as I reck-

I oned

[ 66 ]

dttt& tile whole length of the Colours not to be the whole-
length of the SpeCtrum, hot the length of its rectilinear
fides, fo completing the Semicirlar Ends into Circles, when
either of the obferved Colours fell within thofe Circles, I
meafured the diftance of that Colour from the End of the
SpeCtrum, and fubduCling half the diftance from the mea¬
fured diftance of the Colours, I took the remainder for
their corrected diftance 3 and in thefe Obfervations fet
down this corrected diftance for the difference of their di-
fiances from the Lens. For as the length of the rectilinear
fides of the SpeCtrum would be the whole length of all the
Colours, were the Circles of which ( as we (hewed) that
SpeCtrum confifts contracted and reduced to Fhyfical
Points, fo in that Cafe this corrected diftance would be the
real diftance of the obferved Colours.

When therefore I further obferved the deepeft fenfible Red,
and that Blue whofe corrected diftance from it w^as ^ parts
of the length of the rectilinear fides of the SpeCtrum, the
difference of the diftances of their Foci from the Lens was
about 3 ^ Inches, and as 7-to 12 fo is 3^ to 5 j.

When I obferved the deepeft fenfible Red, and that Indi¬
go whofe corrected diftance w^as ^ or ~ of the length of the
rectilinear fides of the SpeCtrum, the difference of the di¬
ftances of their Foci from the Lens, was about 3 ^Inches,,
and as 2 to 3 fo is 3 *-to 5b

When I obferved the deepeft fenfible Red, and that deep-
Indigo whofe corrected diftance from one another was ^ or
1 of the length of the rectilinear fides of the SpeCtum, the
difference of the diftances of their Foci from the Lens was
about 4 Inches 5 and as 3 to 4 fo is 4 to 5 |.

When I obferved the deepeft fenfible Red, and that part
of the Violet next the Indigo whofe corrected diftance from
Red was or £ of the length of the rectilinear fides of

the.

r.«7]

the Spe&rum, the difference of the diftances of their Foci
from the Lens was about 4^ Inches 5 and as 5 to 6 , fo is
4^ to 57. For fometimes when the Lens was advantage*
oufly placed, fo that its Axis relpe&ed the Blue, and all
things elfe were well ordered, and the Sun flione clear, and
I held my Eye very near to the Paper on which the Lens
call the Species of the Lines, I could fee pretty diftinctly
the Species of thofe Lines by that part of the Violet which
was next the Indigo 3 and fometimes I could fee them by
above half the Violet. For in making thefe Experiments
I had obferved, that the Species of thofe Colours only ap¬
peared diftinct which were in or near the Axis of the Lens:
So that if the Blue or Indigo were in the Axis, I could fee
their Species diftinctly 3 and then the Red appeared much
lefs diftinct than before. Wherefore I contrived to make
the Spectrum of Colours fhorter than before, fo that both
its Ends might be nearer to the Axis of the Lens. And
now its length was about z[ Inches and breadth about l -or
\ of an Inch. Alfo inftead of the black Lines on which the
Spectrum was caft, I made one black Line broader than
thofe, that I might fee its Species more eafily 3 and this
Line I divided by Abort crofs Lines into equal Parts, for
meafuringthe diftancesof the obferved Colours. And now
I could fometimes fee the Species of this Line with its divi-
fions almoft as tar as the Centers of the Semicircular Violet
End of the Spectrum, and made thefe further Observations.

When I obferved the deepeft fallible Red, and that part
of the Violet whofe corrected diftance from it was about
| Parts of the rectilinear fidesof the Speftrum the difference
of the diftances of the Foci of thofe Colours from the Lens,
was one time 4^, another time 4J, another time 4I, Inches,
and as 8 to 9, fo are 4*, 4*, 4$, to 5 * 5^5^ refpedively.

I 2 When

[ 68 ]

When I obferved the deepeft fenfible Red, and deepeft
fenfible Violet, (the Corrected diftance of which Colours
when all things were ordered to the beft advantage, and the
Sun fihone very clear, was about £ or ^ parts of the length
of the rectilinear fides of the coloured Spectrum, ) I found
the difference of the diftances of their Foci from the Lens
fbmetimes 4^ fometimes 5^ and for the mod part 5 Inches
or thereabouts : and as 11 to 1 2 or 15 to 26, fo is five
Inches to 5 or 5 l Inches.

And by this progreffion of Experiments I fatisfied my
felf, that had the light at the very Ends of the Spectrum been
ftrong enough to make the Species of the black Lines ap¬
pear plainly on the Paper, the Focus of the deepeft Vio¬
let would have been found nearer to the Lens, than the Fo¬
cus of the deepeft Red, by about 5 *- Inches at leaft. And
this is a further Evidence, that the Sines of Incidence and
RefraCtion of the feveral forts of Rays, hold the fame pro¬
portion to one another in the fmalleft Refractions whicE
they do in the greateft.

My progrefs in making this nice and troublefome Expe¬
riment I have fet down more at large, that they that fhali
try it after me may be aware of the CircumfpeCtion re-
quifite to make it fucceed well. And if they cannot make
it fucceed fo well as I did, they may notwithftanding coi¬
led by the Proportion of the diftance of the Colours in the
SpeCtrum, to the difference of the diftances of their Foci
from the Lens, what would be the fuccefs in the more di-
ftant Colours by a better Trial. And yet if they ufe a
broader Lens than I did, and fix it to a long {freight Staff
by means of which it may be readily and truly directed to
the Colour whofe Focus is defired, I queftion not but the
Experiment will fucceed better with them than it did with
me,. For I directed the Axis as nearly as I could to the

middle

[<?9J

middle of the Colours, and then the faint Ends of the
Spedmm being remote from the Axis, caft their Species lefs
diftindly on the Paper than they would have done had the
Axis been fucceffively direded to them.

Now by what has been faid its certain, that the Rays
which differ in refrangibility do not converge to the fame
Focus, but if they flow from a lucid point, as far from
the Lens on one fide as their Foci are one the other, the
Focus of the moft refrangible Rays fliall be nearer to the
Lens than that of the leaft refrangible, by above the four¬
teenth part of the whole diftance: and if they flo w from a lu¬
cid point, fo very remote from the Lens that before their
Incidence they may be accounted Parallel, the Focus of the
moft refrangible Rays fliall be nearer to the Lens than the
Focus of the leaft refrangible, by about the 27th or 28th part
of their whole diftance from it. And the Diameter of the
Circle in the "middle fpace between thofe two Foci which
they illuminate when they fall there on any Plane, perpen¬
dicular to the Axis (which Circle is the leaft into which
they can all be gathered) is about the 55th part of the Dia¬
meter of the aperture of the Glafs. So that tis a wonder
that Telefcopes reprefent Objeds fo diftind as they do. But
were all the Rays of Light equally refrangible, the Error
arifing only from the fphericalnefs of the Figures of Glades
would be many hundred times Ids. For if the Objed-
Glafs of a Telefcope be Plano-convex, and the Plane fide
be turned towards the Objed, and the Diameter of the
Sphere whereof this Glafs is a fegment,be called D, and the
Semidiameter of the aperture of the Glafs be called S, and
the Sine of Incidence out of Glafs into Air, be to the Sine of
Refradion as I to R: the Rays which come Parallel to the
Axis of the Glafs, fliall in the Place where the linage of the
Objed is moft diftindly made, be fcatcered all over a little

Circle-

[ 7 °]

Circle whofe Diameter is j * very nearly, as I ga~

ther by computing the Errors of the Rays by the method
of infinite Series, and rejecting the Terms whofe quanti-
tities are inconfiderable. As for inftance, if the Sine of In¬
cidence I, be to the Sine of Refraftion R, as 20 to 51, and
if D the Diameter of the Sphere to which the Convex fide
of the Glafs is ground, be 100 Feet or 1200 Inches, and
S the Semidiameter of the aperture be two Inches, the

Diameter of the little Circle ( that is ) will be

31 x 8

20 X 1200 X 1200

2 X

( or ^— ) parts of an Inch. But the

Diameter of the little Circle through which thefe Rays are
fcattered by unequal refrangibility, will be about the 55 th
part of the aperture of the Objeft-Glafs which here is four
Inches. And therefore the Error arifing from the fpherical
figure of the Glafs, is to the Error arifing from the diffe¬
rent Refrangibility of the Rays, as TS ^ ) to ± that is as 1
to 8151 : and therefore being in Comparifon fo very little,
deferves not to be confidered.

But you will fay, if the Errors caufed by the different re¬
frangibility be fo very great, how comes it to pals that Ob-
|efts appear through Telefcopes fo diftinft as they do ? I an-
fw T er, ? tis becaufe the erring Rays are not fcattered uniform¬
ly over all that circular fpace, but collected infinitely more
denfely in the Center than in any other part of the Circle,
and in the way from the Center to the Circumference grow
continually rarer and rarer, fo as at the Circumference to
become infinitely rare5 and by reafon of their rarity are
not ftrong enough to bevifible, unleis in the Center and ve¬
ry near it. Let ADE reprefent one of thofe Circles de-
fcribed with the Center C and Semidiameter AC, and let
BFG be a fmaller Circle concentric to the former, cutting

• with

[ 7 1 ]

with its Circumference the Diameter AC in B, and befect
AC in N, and by my reckoning the denfity of the Light
in any place B will be to its denfity in N, as AB to BC$
and the whole Light within the leffer Circle BFG, will be
to the whole Light within the greater AED, as the Excefs of
the Square of AC above the Square of AB, is to the Square
of AC. As if BC be the fifth part of AC, the Light will be..
four times denfer in B than in N, and the whole Light with-
in the lefs Circle,will be to the whole Light within the grea¬
ter, as nine to twenty five. Whence its evident that the
Light within the lefs Circle, muftftrike the fenfe much more
ftrongly, than that faint and dilated light round about be¬
tween-it and the Circumference of the greater.

But its further to be noted, that the mod luminous of
the prifmatick Colours are the Yellow and Orange. Thefe
affedt the Senfes more ftrongly than all the reft together, and v
next to thefe in fifength are the Red and Green. The Blue
compared with thefe is a faint and dark Colour, and the In¬
digo and Violet are much darker and fainter, fo that thefe
compared with the ftronger Colours are little to be regard¬
ed. The Images of Objects are therefore to be placed, not
in the Focus of the mean refrangible Rays which are in the:
confine of Green and Blue, but in the Focus of thofe Rays
which are in the middle of the Orange and Yellow 5 there
where the Colour is mod luminous and fulgent, that is in
the brighteft Yellow, that Yellow which inclines., more to **
Orange than to Green. And by the Refradtion of thefe
Rays ( whofe Sines of Incidence and Refraction in Glafs
are as 17 and 1 1 ) the Refradtion of Glafs and Cryftal for ::
optical ufes is to be meafured., Let us therefore place, the :
Image of the Objedt in the Focus of thefe Rays, and all the
Yellow and Orange will fall within a Circle, whofe Dia- -
meter is. about. the z 5 oth part of the Diameter of the aper¬
ture

[ y 2]

ture of the Glafs. And if you add the brighter half of the
Red, (that half which is next the Orange, and the brighter
half of the Green, (that half which is next the Yellow,} a-
bout three fifth parts of the Light of thefe two Colours will
fall within the fame Circle,and two fifth parts will fall with¬
out it round about 5 and that which falls without will be
Ipread through almofl: as much more Ipace as that which
falls within, and fo in the grofs be almofl: three times ra¬
rer. Of the other half of the Red and Green, (that is of
the deep dark Red and Willow Green ) about one quarter
will fall within this Circle, and three quarters without, and
that which falls without will be fpread through about four
or five times more fpace than that which fall within 3 and fo
in the grofs be rarer, and if compared with the whole Light
within it, willbe about 25 times rarer than all that taken in
the grofs 5 or rather more than 3 o or 40 times rarer, be-
caufe the deep red in the end of the Spedrum of Colours
made by a Prifm is very thin and rare,and the Willow Green
is fomething rarer than the Orange and Yellow. The Light
of thefe Colours therefore bring fo very much rarer than that
within the Circle, will fcarce affed the Senfe efpecially fince
the deep Red and Willow Green of this Light, are much
darker Colours then the reft. And for the fame reafon the
Blue and Violet being much darker Colours than thefe, and
much more tariffed, may be neglected. For the denfe and
bright Light of the Circle, will obfcure the rare and weak
Light of thefe dark Colours round about it, and render them
almofl: infenfible. The fenfible Image of a lucid point is
therefore fcarce broader than a Circle whofe Diameter is
the 250th part of the diameter of the aperture of the Object
Glafs of a good Telefcope, or not much broader, if you
except a faint and dark mifty light round about it, which
aSpedator will fcarce regard. And therefore in a Telefcope

whofe

1 73 :i

whofe aperture is four Inches, and length an hundred Feet,
it exceeds not i" 45', or 3”. And in a Telefcope whofe
aperture is two Inches, and length 20 or 30 Feet, it may
be 5 " or 6 " and fcarce above. And this Anfwers well to
Experience : For fome Aftronomers have found the Dia-
meters of the fixt Stars, in Telefcopes of between twenty
and fixty Feet in length, to "be about 4” or 5" or at mol
6 in Diameter. But if the Eye-Glafs be tingled faintly
with the fmoke of a Lamp or Torch, to obfcure the Light
of the Star, the fainter Light in the circumference of the
Star ceafes to be vifible, and the Star (if the Glafs be fuffici-
ently foiled with fmoke) appears fomething more like a Ma¬
thematical Point. And for the fame reafon, the enormous
part of the Light in the Circumference of every lucid Point
ought to be lefs difcernable in flhorter Telefcopes than in
longer, becaufe the fihorter tranfmit lefs Light to the Eye.

Now if we fuppofe the fenfible Image of a lucid point,
to be even 250 times narrower than the aperture of the
Glafs: yet were it not for the different refrangibility of the
Rays, its breadth in an 100 Foot Telefcope whofe aperture
is 4 Inches would be but parts of an Inch, as is ma-

nifeft by the foregoing Computation. And therefore in
this Cafe the greateft Errors arifing from the fpherical Figure
of the Glafs, would be to the greateft fenfible Errors ari¬
fing from the different refrangibility of the Rays as ~~
to — at moft, that is only as 1 to 18 2 6 . And this fuffi-
ciently {hews that it is not the fpherical Figures of Glaffes
but the different refrangibility of the Rays which hinders the
perfection of Telefcopes,

There is another Argument by which it may appear that
the different refrangibility of Rays, is the true Caufe of the
imperfection of Telefcopes. For the Errors of the Rays
arifing from the fpherical Figures of ObjeCt-Glaffes, are as

K the

the Cubes of the apertures of the ObjeCLGlafles • and thence *
to make Telefcopes of various lengths, magnify with equal
diftinCtnefs, the apertures of the Objed> Glafles, and the
Charges or magnifying Powers, ought to be as the Cubes of
the fcjuare Roots of their lengths 5 which doth not anfwer
to Experience. But the errors of the Rays ariflng from
the different refrangibility, are as the apertures of the Ob¬
ject- Glafles,. and thence to make Telefcopes of various
lengths, magnify with equal diftinCtnefs, their apertures and
charges ought to be as the fquare Roots of their lengths 5
and this anfwers to experience as is well known. For in-
ftance, a Telefcope of 64 Feet in length, with an aperture
of 1- Inches, magnifies about 1 20 times, with as much dif¬
tinCtnefs as one of a Foot in length, with ~ of an Inch aper¬
ture, magnifies 15 times.

Now were it not for this different refrangibility of Rays,
Telefcopes might be brought to a greater Perfection than
we have yet defcribed, by compofing the ObjeCt-Glafs of
two Glafles with Water between them. Let ADFC repre¬
sent the ObjeCt-Glafs compofed of two Glafles ABED and
and BEFC, alike convex on the outfides AGD and CFIF,
and alike concave on the infides BME, BNE, with Water
in the concavity BMEN. Let the Sine of Incidence out of
Glafs into Air be as I to R and out of Water into Air as K

to R, and by confequence out of Glafs into Water, as I to
K : and let the Diameter of the Sphere to which the convex
fides AGD and CHF are ground be D, and the Diameter
of the Sphere to which the concave fides BME and BNE
are ground be to D, as the Cube Root of KK—KI to the
Cube Root of RK—RI: and the Refractions on the con¬
cave fides of the Glafles, will very much correct the Errors
of the Refractions on the convex fides, fo far as they arife
from the fphericalnefs of the Figure. And by this means

might. _

[ 7 $]

might Telefcopes be brought to fufficient perfection, were it
not for the different refrangibility of feveralforsof Rays. But
by reafon of this different refrangibility, I do not yet fee any
other means of improving Telefcopes by Refractions alone
than that of increafing their lengths, for which end the late
contrivance of Hugenius feems well accommodated. For
very long Tubes are cumberfome, and fearce to be readily
managed, and by reafon of their length are very apt to
bend, and fhake by bending fo as to caufe a continual
trembling in the ObjeCts, whereby it becomes difficult to
fee them diftin&ly : whereas by his contrivance the GlafTes
are readily manageable, and the ObjeCtiGlafs being fixt up¬
on a ftrong upright Polebecomes more fteddy.

Seeing therefore the improvement of Telefcopes of given
lengths by Refractions is defperate 3 I contrived heretofore a
PerfpeCtive by reflexion, ufing inftead of an ObjeCt Glafs
a concave Metal. The diameter of the Sphere to which
the Metal was ground concave was about 25 Englifli Inches,
and by confequence the length of the Inftrument about fix
Inches and a quarter. The Eye-Glafs was plano-convex,
and the Diameter of the Sphere to which the convex fide was
ground was about L of an Inch, or a little lefs, and by con¬
fequence it magnified between 50 and 40 times. By ano¬
ther way of meafuring I found that it magnified about
35 times. The Concave Metal bore an aperture of an Inch
and a third part 3 but the aperture was limited not by an
opake Circle, covering the Limb of the Metal round about,
but by an opake circle placed between the Eye-Glafs and the *
Eye, and perforated in the middle with a little round hole
for the Rays to pafs through to the Eye. For this Circle
by being placed here, ftopt much of the erroneous Light,
which otherwife would have difturbed the Villon. By com¬
paring it with a pretty good PerfpeCtive of four Feet in

K 2 length,

[ 7 6 ]

length, made with a concave Eye-Glafs, L could read at a
greater diftance with my own Inftrument than with the.
Glafs. Yet Objects appeared much darker in it than in the
Glafs, and that partly becaufe more Light was loft by re¬
flexion in the Metal, then by refraction in the Glafs, and
partly becaufe my Inftrument was overcharged. Had it.
magnified but 50 or 2 5 times it would have made the Object
appear more brisk and pleafant. Two of thefelmade about
16 Years ago, and have one of them ftill by me by which
L ean prove the truth of what I write. Yet it is not fo good
as at the firft. For the concave has been divers times tar-
niftied and cleared again, by rubbing it with very foft Lea¬
ther. When I made thefe, an Artift in London undertook
to imitate it 3 but ufing another way of polifhing them
than I did, he fell much-Abort of what I had attained to,
as I afterwards underftood by difeourfing the under-Work¬
man he had imployed. The Polifih I ufed was on this man¬
ner. I had two round Copper Plates each fix Inches in
Diameter, the one convex the other concave, ground ve¬
ry true to one another. On the convex I ground the. Oh-
jeCl-Metal or concave which was to hepoliflbd, till it had.
taken the Figure of the convex and was ready for a Polifh.
Then I pitched over the convex very thinly, by droppings
melted pitch upon it and warming it to keep the pitch
foft, whilft I ground it with the concave Copper wetted to
make it fpread evenly all over the convex. Thus by work¬
ing it well I made it as thin as a Groat, and after the .con¬
vex was cold I ground it again to give it as true a Figure as,
I could. Then I took.Putty which I had made very fine
by wafhing it from all its grolfer Particles,, and laying a lit¬
tle of this upon the pitch, I ground it upon the Pitch with
the concave Copper till it had done making a noife 3 and
then upon the pitch I ground the Objeft-Metal with a brisk
■ • " *' " * .Motion,

C 77 ' }

Motion, for about two or three Minutes of time, leaning
hard upon it,. Then I put frefh Putty upon the Pitch and
ground it again till it had done making a noife, and after¬
wards ground the Gbjed Metal upon it as before. And
this Work I repeated till the Metal was poliflied, grinding
it the laft time with all my ftrength for a good while toge¬
ther, and frequently breathing upon the Pitch to keep it
moift without laying on any more frefli Putty. The Ob-
jed-Metal was two Inches broad and about one third part
of an Inch thick, to keep it from bending. I had two of
thefe Metals, and when I had polifihed them both I tried
which was bell, and ground the other again to fee if I could
make it better than that which I kept. And thus by many
Trials I learnt the way of polifhing, till I made thofe two
refleding Peipedives I fpake of above. For this Art of
poliihing will he better learnt by repeated Pradice than by
my defcription. Before I ground the Qbjed MetaTon the
Pitch, I always ground the Putty on it with the concave
Copper till it had done making a noife, becaufe if the Par¬
ticles of the Putty were not by this means made to ftick
faft in the Pitch, they would by rolling up and down grate
and fret the Objed Metal and fill it full of little holes.

But becaufe Metal is more difficult to polifih than Glafs
and is afterwards very apt to be fpoiled by tamifhing ? and
refleds not fo much Light as Glafs quick-filvered over does:

I would propound touleinfteadof theMetal, a Glafs ground
concave on the forefide, and as much convex on the back-
fide, and quicksilvered over on the convex fide. The Glafs s
muft be every where of the fame thicknefs exadly. Other-
wife it will make Objeds look coloured and indiftind. By
fuch a Glafs I tried about five or fix Years ago to make
a refleding Telefcope of four Feet in length to magnify a- -
bbut 150 times, and I fatisfied my felf that there wants no--

•vj?

tiling but a good Artift to bring the defign to Perfection.
For the Glafs being wrought by one of our London Artifts
after fuch a manner as they grind Glaffes for Telefcopes.,
tho it feemed as well wrought as the ObjeCi Glaffes ufe to
be, yet when it was quick-filvered, the reflexion difcovered
innumerable Inequalities all over the Glafs. And by reafon
of thefe Inequalities, ObjeCts appeared indiftinCt in this In-
ftrument. For the Errors of reflected Rays caufed by any
Inequality of the Glafs, are about fix times greater than the
Errors of refraCled Rays caufed by the like Inequalities. Yet
>by this Experiment I fatisfied my felf that the reflexion on
the concave fide of the Glafs, which I feared would difturb
the vifion,didno fenfibleprejudice to it, and by confequence
that nothing is wanting to perfect thefe Telefcopes, but
good Workmen who can grind and polilh Glaffes truly fphe-
rical. An ObjeCt-Glafs of a fourteen Foot Telefcope, made
by one of our London Artificers, I once mended confidera-
bly, by grinding it on Pitch with Putty, and leaning ve¬
ry eafily on it in the grinding, left the Putty flhould fcratch
it. Whether this way may not do well enough for poliflh-
ing thefe reflecting Glaffes, I have not yet tried. But he
that flhall try either this or any other way of poliflhing which
he may think better, may do well to make his Glaffes rea¬
dy for poliflhing by grinding them without that violence,
wherevuth our London Workmen prefs their Glaffes in grind¬
ing. For by fuch violent preffure, Glaffes are apt to bend
a little in the grinding, and fuch bending will certainly Ip oil
their Figure. To recommend therefore the confideration
of thefe reflecting Glaffes, to fuch Artifts as are curious in
figuring Glaffes, I flhall defcribe this Optical Inftrument in
, the following Propofition.

PROP.

[ 79 ]

PROP. VII. Prob. II.

To Jhorlen Tele/copes.

T E T ABDC reprefent a Glafs fpherically concave on
the forefide AB, 1 and as much convex on the back-
fide CD, fo that it be every where of an equal thicknefs. Let
it not be thicker on one fide than on the other, left it make
Objects appear coloured and indiftindt, and let it be very
truly wrought and quick-filveredoveron the backfide 3 and
fet in the Tube VXYZ which muft be very black within.
Let EFG reprefent a Prifm of Glafs or Cryftal placed near
the other end of the Tube, in the middle of it, by means of
a handle of Brafs or Iron FGK, to the end of which made
flat it is cemented. Let this Prifm be rectangular at E, and
let the other two Angles at F and G be accurately equal to
each other, and by confequence equal to half right ones, and
let the plane tides FE and GE be fquare, and by confe¬
quence the third fideFG a rectangular parallelogram, whofe
length is to its breath in a fubduplicate proportion of two
to one. Let it be fo placed in the Tube, that the Axis of
the Speculum may pafs through the middle of the fquare
fide EF perpendicularly, and by confequence through the
middle of the fide F G at an Angle of 45 degrees, and let the
fide EF be turned towards the Speculum, and the diftance -
of this Prifm from the Speculum be fuch that the Rays of the
light PQ, RS, ScCi which are incident upon the Speculum in *
Lines Parallel to the Axis thereof, may enter the Prifm at
the fide EF, and be reflected by the fide F G, and thence
go out of it through the fide GE, to the point T which
muft be the common Focus of the Speculum ABDC, and of
a Plano-convex Eye- Glafs FI, through which thofe Rays
muft. pafs to the Eye. . And let the Rays at their corning

out

[ 8o ]

Out of the Glafs pafs through a fmall round hole, or aper¬
ture made in a little Plate of Lead, Brafs, or Silver, where¬
with the Glafs is to be covered, which hole muff be no
bigger than is neceiTary for light enough to pafs through.
For fo it will render the Objedt diftindt, the Plate in which
"tis made intercepting all the erroneous part of the Light
which comes from the Verges of the Speculum AB. Such
an Inftrument well made if it be 6 Foot long, ( reckoning
the length from the Speculum to the Prifm, and thence to
the Focus T) will bear an aperture of 6 Inches at the Spe¬
culum, and magnify between two and three hundred times.
But the hole H here limits the aperture with more advan¬
tage, then if the aperture was placed at the Speculum. If
the Inftrument be made longer or fihorter, the aperture mu ft
be in proportion as the Cube of the fquare Root of the
length, and the magnifying as the aperture. But its con¬
venient that the Speculum be an Inch or two broader than
the aperture at the leaft, and that the Glafs of the Speculum
be thick, that it bend not in the working. The Prifm EFG
niuft be no bigger than is neceflary, and its back fide FG
muft not be quick-filvered over. For without quick-filver
it will refledt all the Light incident on it from the Speculum.

In this Inftrument the Objedt will be inverted, but may
be eredted by making the fquare fides EF and EG of the
Prifm EFG not plane but fpherically convex, that the Rays
may crofs as well before they come at it as afterwards be¬
tween it and the Eye-Glafs. If it be defired that the Inftru¬
ment bear a larger aperture, that may be alfo done by com-
pofing the Speculum of two Glaffes with Water between
them.

THE

Book I. Plate I. Part i.

Fig. y.

Book I. Plate I. Parti.

[8i]

THE

FIRST BOOK

O F

O P T I C K s.

PAR T II.

PROP. I, THEOR. I.

1 he Phenomena of Colours in ref ratted or refietted Light
are not caufed by new modifications of the Light varioufi \\
ly imfrejl , according to the variom terminations of the
Light and Shadow .

The Proof by Experiments.

EX PER. I.

F O R if the Sun fhine into a very dark Chamber pV %
through an oblong Hole F, whofe breadth is the °*
lixth or eighth part'of an Inch, or fomething lefs ; and
his Beam F H do afterwards pafs firft through a very
large Prifm ABC, diftant about oo Feet from the

L • Hole,

'[82]

Hole, and parallel to it, and then (with its white part)
through an oblong Hole H, whole breadth is about
the fortieth or fixtieth part of an Inch, and which is
made in; a black opake Body G I, and placed at the
diftance of two or three Feet from the Prifm, in a pa¬
rallel lituation both to the Prifm and to the former
Hole, and if this white Light thus tranfmitted through
the Hole H, fall afterwards upon a white Paper PL
placed after that Hole H, at the diftance of three or
four Feet from it, and there paint the ufual Colours of
the Prifm, fuppofe red at t, yellow at s, green at r,
blue at q, and violet at p ; you may with an iron Wire,
or any fuch like (lender opake Body, whole breadth is
about the tenth part of an Inch, by intercepting the rays
at k, 1, m, n or o, take away any one of the Colours*
at t, s, r, q or p, whilft the other Colours remain up¬
on the Paper as before; or with an obftacle fomething
bigger you may take away any two, or three, or four Co¬
lours together, the reft remaining: So that any one of
the Colours as well as violet may become outmoft in
the confine of the fhadow towards p, and any one of
them as well as red may become outmoft in the confine
of the lhadow towards t, and any one of them may alfo
border upon the fhadow made within the Colours by
the obftacle R intercepting fome intermediate part of
the Light; and, laftly, any one of them by * being
left alone may border upon the fhadow on either hand.
All the Colours have themfelves indifferently to any
confines of fhadow, and therefore the differences of thefe
Colours from one another, do not arife from the diffe¬
rent confines of fhadow, whereby Light is varioufly
modified as has hitherto been the Option of Philofo-

phers.

r «3 ]

phers. In trying thefe things Lis to be obferved, that
by how much the Holes F and H are narrower, and the
intervals between them, and the Prifm greater, and the
Chamber darker, by fo much the better doth the Ex¬
periment fucceed ; provided the Light be not fo far
diminifhed, but that the Colours at p t be fufficiently
viiible. To procure a Prifm of folid Glafs large enough
for this Experiment will be difficult, and therefore a
prifmatick Veffel muft be made of polifhed Glafs-plates
cemented together, and filled with Water,

EXPEL II.

The Sun’s Light let into a dark Chamber through Fig.
the round Hole F, half an Inch wide, paffed firft through
the Prifm ABC placed at the Hole, and then through
a Lens PT fomething more than four Inches broad, and
about eight Feet diftant from the Prifm,and thence con¬
verged to O the Focus of the Lens diftant from it about
three Feet, and there fell upon a white Paper DE„ If
that Paper was perpendicular to that Light incident up-
on it, as Lis reprefented in the pofture D E, all the Co¬
lours upon it at O appeared white* But if the Paper
being turned about an Axis parallel to the Prifm, be¬
came very much inclined to the Light as Lis reprefeiv
ted in the pofttions de and <h • the fame Light in the
one cafe appeared yellow and red, in the other blue.
Here one and the fame part of the Light in one and the
fame place, according to the various inclinations of the
Paper, appeared in one cafe white, in another yellow
or red, in a third blue, whilft the confine of Light and

L 2 Shadow^

^ 1

.... .. . [ 84]

Shadow, and the refrattions of the Prifm in all thefe
cafes remained the fame.

EX PER. III.

into a dark Chamber through a Hole in the Window
Unit be refratted by a large Prifm ABC, whole re-
flatting Angle C is more than 60 degrees, and fo foon
as it comes out of the Prifm let it fall upon the white
Paper D E glewed upon a ftiff plane, and this Light,
when the Paper is perpendicular to it, as 5 tis reprefen-
ted in D E, will appear perfectly white upon the Paper,
but when the Paper is very much inclined to it in Inch
a manner as to keep always parallel to the Axis of the
Prifm, the whitenefs of the whole Light upon the
Paper will according to the inclination of the Paper
this way, or that way, change either into yellow and
red, as in the pofture de , or into blue and violet, as
in the pofture And if the Light before it fall upon
the Paper be twice refratted the fame way by two pa¬
rallel Prifms, thefe Colours will become the more con-
fpicuous. Here all the middle parts of the broad beam
of white Light which fell upon the Paper, did without
any confine of fhadowWo modify it, become coloured
all over with one uniform Colour, the Colour being al¬
ways the fame in the middle of the Papa' as at the
edges, and this Colour changed according the various
obliquity of the reftetting Paper, without any change
in the refrattions or fhadow, or in the Light which
fell upon the Paper. And therefore thefe Colours are

3. Such another Experiment may be more eafily tried
as follows. Let a broad beam of the Sun’s Light coming

'[ 85 j

to be derived from fome other caufe than the new mo¬
difications of Light by refraftions and Shadows.

If it be asked, What then is their caule? I anfwer,
That the Paper in the pofture de , being more ob¬
lique to the more refrangible rays than to the lefs re¬
frangible ones, is more ftrongly illuminated by the laL
ter than by the former, and therefore the lefs refran¬
gible rays are predominant in the reflected Light. And
wherever they are predominant in any Light they tinge
it with red or yellow, as may in fome meaftire appear by
the firft Proposition of the firft Book,and will more fully
appear hereafter. And the contrary happens in the
pofture of the Paper ^ s , the more refrangible rays be¬
ing then predominant which always tinge Light with ,
blues and violets.

EX PER. IV.

The Colours of Bubbles with which Children play
are various, and change their fttuation varioufly, with¬
out any refpedft to any confine of ftiadow. If fuch a
Bubble be covered with a concave Glafs, to keep it from
being agitated by any wind or motion of the Air, the
Colours will flowly and regularly change their fixa¬
tion, even whilft the Eye, and the Bubble, and all Bo^
dies which emit any Light, or caft any ftiadow, re¬
main unmoved. And therefore their Colours arife from
fome regular caufe which depends not on any confine of
ftiadow. What this caufe is will be fhewed in the next
Book,

[8<S]

To thefe Experiments may be added the tenth Ex¬
periment of the firft Book, where the Sun’s Light in a
dark Room being traje&ed through the parallel fuperfi-
cies of two Prifms tied together in the form of a Paral-
lelopide, became totally of one uniform yellow or red
Colour, at its emerging out of the Prifms. Here, in
the production of thefe Colours, the confine of fhadow
can have nothing to do. For the Light changes from
white to yellow,orange and red fucceffively,without any
alteration of the confine of fhadow: And at both edges of
the emerging Light where the contrary confines of fha¬
dow ought to produce different effeCts, the Colour is
one and the fame, whether it be white, yellow, orange
or red : And in the middle of the emerging Light,
where there is no confine of fhadow at all, the Colour
is the very fame as at the edges, the whole Light at its
very firft emergence being of one uniform Colour, whe¬
ther white, yellow, orange or red, and going on thence
perpetually without any change of Colour, fuch as the
confine of fhadow is vulgarly fuppofed to work in re-
fraCted Light after its emergence. Neither can thefe
Colours arife from any new modifications of the Light
by refraClions, becaufe they change fucceffively from
white to yellow, orange and red, while the refractions
remain the fame, and alfo becaufe the refractions are
made contrary ways by parallel fuperficies which de¬
ft r®y one anothers effeCts. They arife not therefore
from any modifications of Light made by refractions
andfhadows, but have fome other caufe. What that
caufe is *we fhewed above in this tenth Experiment,
.and need not here repeat it.

There

[ 8 7 ]. .

There is yet another material circumfiance of this
Experiment. For this emerging Light being by a third Fig . 22^
Prifm HIK refracted towards the Paper P T, and there Tart 1
painting the ufual Colours of the Prifm-, red, yellow,
green, blue, violet : If thefe Colours arofe from the
refractions of that Prifm modifying the Light, they
would not be in the Light before its incidence on that
Prifm. And yet in that Experiment we found that
when by turning the two firft Prifms about their com¬
mon Axis all the Colours were made to vanifh but the
red; the Light which makes that red being left alone,
appeared of the very fame red Colour before its inci¬
dence on the third Prifm. And in general we find by
other Experiments that when the rays which differ in
refrangibility are feparated from one another, and any
one fort of them is confidered apart, the Colour of the
Light which they compofe cannot be changed by any
refraction or reflexion whatever, as it ought to be were
Colours nothing elfe than modifications of Light caufied v
by refractions, and reflexions, and fhadows. This un-
ehangeablenefs of Colour I am now to defcribe in the
following Propofition.

PROP. IL THEOR.IL

AU homogeneal Light has its proper Colour anfwering to-
its degree of refrangibility, and that Colour cannot be
changed by reflexions and refractions*

In the Experiments of the 4th Propofition of the fir ft
Book, when I had feparated the heterogeneous rays
from one another, the SpeCtrum p t formed by the fepa¬
rated

[ 88 ]

rated rays, did in the progrefs from its end p, on which
the molt refrangible rays fell, unto its other end t, on
which the leaft refrangible rays fell, appear tinged with
this Series of Colours, violet, indico, blue, green, yel¬
low, orange, red, together with all their intermediate
degrees in a continual fucceffion perpetually varying:
So that there appeared as many degrees of Colours, as
there were forts of rays differing in refrangibility.

EX PER. V.

Now that thefe Colours could not be changed by re¬
fraction, I knew by refracting with a Prifrn fometimes
one very little part of this Light, fometimes another
very little part, as is defcribed in the iath Experiment
of the firft Book. For by this refraCtion the Colour of
the Light was never changed in the leaft. If any part
of the red Light was refraCted, it remained totally of
the fame red Colour as before. No orange, no yel¬
low, no green, or blue, no other new Colour was pro¬
duced by that refraction. Neither did the Colour any
ways change by repeated refractions, but continued al¬
ways the fame red entirely as at firft. The like con-
ftancy and immutability I found alfo in the blue, green,
and other Colours. So alfo if I looked through a Prifrn
upon any body illuminated with any part of this homo-
geneal Light, as in the 14th Experiment of the firft
Book is defcribed; I could not perceive any new Co¬
lour generated this way. All Bodies illuminated with
compound Light appear through Prifms confufed ( as
was faid above) and tinged with various new Colours,
but thofe illuminated with homogeneal Light appeared

through

[» 9 ]

through Prifins neither lefs diftinfit, nor otherwife co¬
loured, than when viewed with the naked Eyes. Their
Colours were not in the leaft changed by the refraction
of the interpofed Prifm. I fpeak here of a fenfible
change of Colour : For the Light which I here call ho¬
mogeneal, being not abfolutely homogeneal, there ought
to arife fome little change of Colour from its heteroge¬
neity. But if that heterogeneity was fo little as it might
be made, by the laid Experiments of the fourth Fropo-
fition, that change was not fenfible, and therefore, in
Experiments where fenfe is judge, ought, to be accoun¬
ted none at alb

EX PER. VI.

And as thefe Colours were not changeable by refra¬
ctions, fo neither were they by reflexions. For all
white, grey, red, yellow, green, blue, violet Bodies, as
Paper, Allies, red‘Lead, Orpiment, Indico, Bile, Gold,
Silver, Copper, Grafs, blue Flowers, Violets, Bubbles
of Water tinged with various Colours, Peacock's Fea¬
thers, the tinCture of Lignum Neghriticum , and fuch
like, in red homogeneal Light appeared totally red, in
blue Light totally blue, in green Light totally green,
and to of other Colours, In the homogeneal Light of
of any Colour they all appeared totally of that fame
Colour, with this onlf difference, that fome of them
reflected that Light more ftrongly, others more faintly,

I never yet found any Body which by reflecting homo¬
geneal Light could fenfibly change its Colour.

From

[ 90 ]

From all which it is manifeft, that if the Sun’s Light
confifted of but one fort of rays, there would be but
one Colour in the whole World, nor would it be pof-
fible to produce any new Colour by reflexions and re-
fractions, and. by confequence that the variety of Co¬
lours depends upon the competition of Light.

The homogeneai light and rays which appear red T ,
or rather make Objects appear lb, 1 call rubrific •
orred-makng ; thole which make Objects appear
yellow, green, blue and violet, I call yellow-mar¬
king, green-making^ blue-making, violet-making,
and fo of the reft. And if at any time I fpeak of
light and rays as coloured or endued with Co¬
lours, I would be underftood to fpeak not philo-
fophically and properly, but groily, and accor¬
ding to fuch conceptions as vulgar Feople in fee¬
ing all thefe Experiments would be apt to frame. *
For the rays to fpeak properly are not coloured.
In them there is nothing elfe than a. certain power,
and difpolition to ftir up a fenlation of this or that
Colour. For as found in a Bell ori mufical String,
or other founding Body, . is nothing but a trem¬
bling Motion, and in the Air nothing but that
Motion propagated from, the Objed:, and in the .
Senforium ’tis a fenfe of that Motion under the :
form of found ; fo Colours in the Object are no¬
thing but a difpofition to refled this or that fort
of rays more copioufty than the reft; in the rays
they are nothing but their difpofttions to propa¬
gate

Fip: 8

Fig. 12 .

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[?l]

gate this or that Motion into the Senforium, and
in the Senforium they are fenfations of thole Mo¬
tions under the forms of Colours.

PROP. III. PRO EL I.

To define the refrcmgibihty of the feverd forts of homo*
genecd Light anfwering to the feverd Colours .

For determining this Problem I made the following
Experiment.

EXPER, V 1 L

1 When I had caufed the redilinear line fides AF, GM, Fig . 4,
of the Spedrum of Colours made by the Prifm to be
diftindly defined, as in the fifth Experiment of the
firft Book is defcribed, there were found in it all the
homogeneal Colours in the fame order and Situation
one among another as in the Spedrum of fimple Light,
defcribed in the fourth Experiment of that Book. For
the Circles of which the Spedrum of compound Light
FT is compofed, and which in the middle parts of
the Spedrum interfere and are intermixt with one ano¬
ther, are not intermixt in their outmoft parts where
they touch thofe redilinear fides A F and G M. And
therefore in thofe redilinear fides when diftindly defi¬
ned, there is no new Colour generated by refradiom I
obferved alio, that if any where between the two out¬
moft CirclesTMF and PG A a right line, as 5 was
crofs to the Spedrum, fo as at both ends to fall per¬
pendicularly upon its redilinear fides, there appeared

M a one'

0 3

one and the fame Colour and degree of Colour from one
end of this line to the other. 1 delineated therefore in
a Paper the perimeter of the SpeCtrum F APGMT,
and in trying the third Experiment of the firft Book, I
held the Paper fo that the Spectrum might fall upon
this delineated Figure, and agree with it exaCtly, whilft
an Affiftant whofe Eyes for diftinguifhing Colours were
more critical than mine, did by right lines
drawn crofs the SpeCtrum, note the confines of the Co¬
lours that is of theredM*£F of the orange * 7 * of
the yellow 7 g i of the green ^ 0 1 , of the blue >1 * x.e ^
of the indico «and of the violet xGAm, And
this operation being divers times repeated both in the
fame and in federal Papers, I found that the Ob-
fervations agreed well enough with one another, and
that the rectilinear fides M G and F A were by the laid
crofs lines divided after the manner of a mufical Chord.
Let GM be produced to X, that MX may be equal
to GM, and conceive G X, xX, J X, * X, *X, 7X,

M X, to be in proportion to one another, as the num¬
bers 1, L L p b *6’ L and fo to reprefent the.
Chords of the Key, and of a Tone, a third Minor, a
fourth, a fifth, a fixth Major, a feventh, and an eighth
above that Key : And the intervals «y, 7 %..** 9
xx ? all d xG, will be the ipaces which the leveral Co¬
lours (. red, orange, yellow, green, blue, indico, violet )
take up.^

Now thefe- intervals or fpacts fubtending the diffe¬
rences of the refraftions of the rays going to the limits,
of thefe Colours, that is^ to the points M, <*, #,.. h *,/, x, G,
may without any fenfible.. Error be accounted propor¬
tional to the differences of the fines of net raCt ion of.th.ofe.

rays

[ 93 ]. .

rays having one common fine of incidence, and there¬
fore fince the common fine of incidence of the inoft and
leaft refrangible rays out of Glafs into Air was, (by a
method defcribed above ) found in proportion to their
fines of refraction, as 50 to 77 and 78, divide the dif¬
ference between the fines of refraction 77 and 78, as the
line G M is divided by thole intervals, you will have

n, n\y 77 }» 77 j, 1% 77 a 77^,78,, the fines of
refraCtion of thofe rays out of Glais into. Air , their

common fine of incidence being 50. So then the fines
of the incidences of all the red-making rays out of
Glafs into Air, were to the fines- of their refraCtions,
not greater than 50 to 77, nor lefs than 50 to77-, but
varied from one another according to all interme¬
diate Proportions. And the fines of the incidences
of the green-making rays were to the fines of
their refraCtions in all proportions from that of 50
to 77- 1 , unto that of 50 to 77-f And by the like limits
above-mentioned were the refraCtions of the rays be¬
longing to the reft of the Colours defined, the fines of
the red-making;rays extending from 77 to 77^ thofe
of the orange-making from 77 \ to , thofe of the yel¬
low-making from 77~ to 77thole of the green-making
from 77^ to 7 7G thofe of the blue-making from jj~ to
77“, thofe of the indico-making from 77- to 77*, and
thofe of the violet from 77 z 9 to 78,.

Thefe are the Laws of the refraCtions made out of
Glafs into Air, and thence by the three Axioms of the
firft Book the Laws of the refraCtions made out of Air.
into Glafs are eafily derived.

EXPER.

[ 9 +]

EXPER. VIII.

I found moreover that when Light goes out of Air
through feveral contiguous refracting Mediums as
through Water and Glafs, and thence goes out again
into Air, whether the refracting fuperficies be parallel
or inclined to one another, that Light as often as by
contrary refraCtions "tis fo corrected, that it emergeth
in lines parallel to thofe in which it was incident,
continues ever after to be white. But if the emer¬
gent rays be inclined to the incident, the whitenefs of
the emerging Light will by degrees in paffing on from
the place of emergence, become tinged in its edges with
Colours. This I tryed by refraCting Light with Prifms
of Glafs within a prifmatick Veffel of Water. Now thofe
Colours argue a diverging and reparation of the hetero¬
geneous rays from one another by means of their un¬
equal refraCtions, as in what follows will more fully
appear. And, on the contrary, the permanent white¬
nefs argues, that in like incidences of the rays there is
no fuch reparation of the emerging rays, and by confe-
quence no inequality of their whole refraCtions. Whence
1 feem to gether the two following Theorems.

i. The Exceffes of the fines of refraCtion of feveral
forts of rays above their common fine of incidence when
the refraCtions are made out of divers denfer mediums
immediately into one and the fame rarer medium, are
do one another in a given Proportion

2, The

[ 95 ]

a. The Proportion of the fine of incidence to the line
of refraction of one and the fame fort of rays out of one
medium into another, is compofed of the Proportion of
the fine of incidence to the fine of refraction out of the
firft medium into any third medium, and of the Pro¬
portion of the line of incidence to the fine of refraCtioiv
out of that third medium into the fecond medium*

By the firft Theorem the refractions of the rays of
every fort made out of any medium into Air are known ,
by having the refraction of the rays of any one fort. As
for inftance, if the refractions of the rays of every fort
out of Rain-water into Air be defired, let the common
line of incidence out of Glafs into Air be fubduCted
from the fines of refraCtion, and the Exceffes will be
ay, ay£,-.ayj, if-, ij[, if- r ,if 9y i%. Suppofe now
that the fine of incidence of the leaft refrangible rays be~
to their fine of refraction out of Rain-water into Air as
three to four, and fay as i the difference of thofe fines
is to 3 the fine of incidence, fo is ay the leaft of the.
Exceffes above-mentioned to a fourth number 81 j and
8.1-will be the common fign of incidence out of Rain¬
water into Air, to which fine if you add all the above-
mentioned Exceffes you will have the defired fines of
the refractions 108., io8i? io8f, io8i, io8t, io8f,

I C89 ? IO9.

By the latter Theorem the refraCtion out of., one me¬
dium into another is gathered as often as you. have
the refractions out of them both into any third medium.
As if the fine of incidence of any ray out of Glafs into
Air be to its fine of refraCtion as ao to 31, and the fine
of incidence of the fame ray out of Air into Water, be

[ 9 ^ ]

to its fine of refra£tion as four to three; the fine of
incidence of that ray out of Glafs into Water will be to
its fine of refraction as ao to 31 and 4 to 3 joyntly, that
is, as the FaCtum of 20 and 4 to the Falfum of 31 and
3, or as 80 to 93.

And thefe Theorems being admitted into Opticks,
there would be fcope enough of handling that Science
voluminoufly after a new manner; not only by teaching
thofe things which tend to the perfection of vifion, but
alfo by determining mathematically all kinds of Pheno¬
mena of Colours which could be produced by refra¬
ctions. For to do this, there is nothing elfe requifite
than to find out the reparations of heterogeneous rays,
and their various mixtures and proportions in every
mixture. By this way of arguing I invented almoft
all the Phenomena defcribed in thefe Books, befide fome
others lefs neceflary to the Argument ; and by the
fuccefles I met with in die tryals, I dare promile, that
to him who (hall argue truly, and then try all things
with good Glaffes and fufficient circumfpeCtion, the
expefted event will not be wanting. But he is firft to
know what Colours will arife from any others mixt in
any affigned Proportion.

PROP. IV. THE OR. III.

Colours may be produced by comfofition which jhatt be like
to the Colours of homogeneal Light as to the appearance
of Colour , but not as to the immutability of Colour and
conjlitutim of Light, Lind thofe Colours by ho‘w much
they are more compounded by jo much are they lejs f ull
.md inteufe , and by too much compofition they may be

diluted

[ 97 ]

diluted and weakened till they ceaje . ‘There may he

aljo Colours produced by comnojition^ which are not fully
like any oj the Colours oj homogeneal Light .

For a mixture of homogeneal red and yellow com¬
pounds an orange, like in appearance of Colour to that
orange which in the feries of unmixed prifmatick Co¬
lours lies between them; but the Light of one orange
is homogeneal as to refrangibility, that of the other is
heterogeneal, and the Colour of the one, if viewed
through a Prifm, remains unchanged, that of the other
is changed and refolved into its component Colours red
and yellow. And after the fame manner other neigh¬
bouring homogeneal Colours may compound new Co¬
lours, like the intermediate homogeneal ones, as yel¬
low and green, the Colour between them both, and af¬
terwards, if blue be added, there will be made a green
the middle Colour of the three which enter the compo-
lition. For the yellow and blue on either hand,if they are
equal in quantity they draw the intermediate green equal¬
ly towards themfelves in competition, and fo keep it as
it were in equilibrio, that it verge not more to the
yellow on the one hand, than to the blue on the other,
but by their mixt actions remain ftill a middle Colour.
To this mixed green there may be further added
feme red and violet, and yet the green will not prefent-
ly ceafe but only grow lets full and vivid, and by in-
creafing the red and violet it will grow more and more
dilute, until by the prevalence of the added Colours it
be overcome and turned into whitenefs, or feme other
Colour, do if to the Colour of any homogeneal Light,
the Sun's white Light compofed of all forts of rays be

N added.

added, that Colour will not vanifh or change its fpe*
cies but be diluted., and by adding more and more white
it will be diluted more and more perpetually. Lafi>
ly, if red and violet be mingled, there will be generated
according to their various Proportions various Purples,
fuch as are not like in appearance to the Colour of any
homogeneal Light, and of thefe Purples mixt with yel¬
low and blue may be made other new Colours.

PROP. Y. THEOR. IV.

Whitenefs and all grey Colours between white and blacks
may be compounded of Colours , and the whitenefs of the
Suns Light is compounded of all, the primary Colours
mixt in a due proportion ...

The Proof by Experiments ;

; ■

EX PER. IX.

' pig, ^ The Sun Aiming into a dark Chamber through a

° ' little round Hole in the Window fhut, and his Light
being there refrafted by a Prifm to cafi his coloured-
image PT upon the oppofite Wall : I held a white Pa¬
per V to that Image in fuch manner that it might be
illuminated by the coloured Light refected from thence,
and yet not intercept any part of that Light in its paL
fage from the Prifm to the Spedtrum. And I found that
when the Paper was held nearer to any Colour than to
the reft, it appeared of that Colour to which it ap¬
proached neareft ; but when it. was equally or almoft

equally

[ 99 3

equally difrant from all the Colours, fo that it might
be equally illuminated by them all it appeared white.

And in this la ft fituation of the Paper, iffome Colours
were intercepted, the Paper loft its white Colour, and
appeared of the Colour of the reft of the Light which
was not intercepted. So then the Paper was illuminated
with Lights of various Colours, namely, red, yellow,
green, blue and violet, and every part of the Light re¬
tained its proper Colour, until it was incident on the
Paper, and became reflected thence to the Eye ; fo that
if it had been either alone (the reft of the Light being
intercepted) or if it had abounded moft and been pre¬
dominant in the Light reflected from the Paper,it would
have tinged the Paper with its own Colour; and yet be¬
ing mixed with the reft of the Colours in a due propor¬
tion, it made the Paper look white, and therefore by a
eompofition with the reft produced that Colour, The
feveral parts of the coloured Light reflected from the
-Spedrum, whilft they are propagated from thence thro" \
the Air, do perpetually retain their proper Colours,
becaufe wherever they fall upon the Eyes of any Specta¬
tor, they make the feveral parts of the SpeCtrum to
appear under their proper Colours. They retain there¬
fore their proper Colours when they fall upon the Pa¬
per V, and fo by the confufion and perfect mixture of
thofe Colours compound the whitenefs of the Light
reflected from thence,

EX PER, X,

Let that Spectrum or tolar Image P T fall now upon Fig, 6
the .Lens M N above four Inches broad, and about fix

N 2 Feet

[ i oo ]

Feet diftant from the Prifm ABC, and fo figured that
it may caufe the coloured Light which divergeth from
the Prifm to converge and meet again at its Focus G,
about fix or eight Feet diftant from the Lens, and
thereto fall perpendicularly upon a white Paper DE.
And if you move this Paper to and fro, you will per¬
ceive that near the Lens, as at de y the whole folar Image
(fuppofe at pt) will appear upon it intenfly coloured
after the manner above-explained, and that by receding
from the Lens thofe Colours will perpetually come to¬
wards one another, and by mixing more and more di¬
lute one another continually, until at length the Paper
come to the Focus G, where by a perfect mixture they
will wholly vanifh and be converted into whitenefs, the
whole Light appearing now upon the Paper like a little
white Circle, And afterwards by receding further from
the Lens, the rays which before converged will now
crofs one another in the Focus G, and diverge from
thence, and thereby make the Colours to appear again,
but yet in a contrary order; fuppofe at ^, where the
red t is now above which before was below, and the
violet p is below which before was above.

Let us now flop the Paper at the Focus G where
the Light appears totally white and circular, and let us
eonfider its whitenefs. I fay, that this is com poled of
the converging Colours. For it any of thofe Colours
be intercepted at the Lens, the whitenefs will ceafe and
degenerate into that Colour which arifeth from the
compofition of the other Colours which are not inter¬
cepted. And then if the intercepted Colours he let
pais and fall upon that compound Colour, they mix
with it, and by their mixture reftore the whitenefs.

[ 101 ]

So if the violet, blue and green be intercepted, the re¬
gaining yellow, orange and red will compound upon
the Paper an orange, and then if the intercepted Co¬
lours be let pals they will fall upon this compounded
orange, and together with it decompound a white. So
alfo if the red and violet be intercepted, the remaining
yellow, green and blue, will compound a green upon
the Paper, and then the red and violet being let pafs
will fall upon this green, and together with it decom¬
pound a white. And that in this compofition of white
the feveral rays do not fuffer any change in their colori¬
fic qualities by ading upon one another, but are only
mixed, and by a mixture of their Colours produce
white, may further appear by thefe Arguments.

If the Paper be placed beyond the Focus G, fuppofe
at ^, and then the red Colour at the Lens be alternate¬
ly intercepted, and let pafs again, the violet Colour on
the Paper will not fuffer any change thereby, as it ought
to do if the feveral forts of rays aded upon one another
in the Focus G, where they crofs. Neither will the
red upon the Paper be changed by any alternate flop¬
ping, and letting pafs the violet which crofleth it.

And if the Paper be placed at the Focus G, and the
white round Image at G be viewed through the Prilm
HIK, and by the reflation of that Prilm be tranflated
to the place rv, and there appear tinged with various
Colours, namely, the violet at v and red at r, and
others between, and then the red Colour at the Lens be
often ftopt and let pafs by turns, the red at r will ac¬
cordingly difappear and return as often, but the violet
at v will not thereby fuffer any change. And fo by
flopping and letting pafs alternately the blue at the

Lens,

I *02

'Lens, the blue at r will accordingly difappear and re-
turn, without any change made in the red at r. The
red therefore depends on one fort of rays, and the blue
on another fort, which in the Focus G where they are
commixt do not aft on one another. And there is the
fame reafon of the other Colours.

I confidered further, that when the mod refrangible
rays P p, and the leaft refrangible ones T t, are by con¬
verging inclined to one another, the Paper, if held very
oblique to thofe rays in the Focus G, might refleft one
fort of them more copioufly than the other fort, and by
that means the reflected Light would be tinged in that
Focus with the Colour of the predominant rays, pro¬
vided thofe rays feverally retained their Colours or co¬
lorific qualities in the compofition of white made by
them in that Focus. But if they did not retain them
in that white, but became all of them feverally endued
there with a difpoiition to ftrike the fenfe with the pew
ception of white, then they could never lofe their white-
nefs by fuch reflexions. I inclined therefore the Paper
to the rays very obliquely, as in the fecund Experiment
of this Book, that the rnoft refrangible rays might be
more copioufly reflefted than the reft, and the white-
nefs at length changed fucceffively into blue, indico
and violet. Then I inclined it the contrary way, that
the moft refrangible rays might be more copious in the
reflefted Light than the reft, and the whitenefs turned
fucceffively to yellow, orange and red. . .

Laftly, I made an Inftrument X Y in fafhion of a
Comb, whofe Teeth being in number fixtecn were
about an Inch and an half broad, and the intervals of the
Teeth about two Inches wide. Then by interpofing

fuc-

C io 3 ]

iucceffively the Teeth of this Inftrument near the Lens,
I intercepted part of the Colours by the interpofed
Toothy whilft the reft of them went on through the in¬
terval of the Teeth to the Paper D E, and there pain¬
ted a round folar Image. But the Paper I had firft pla¬
ced fo, that the Image might appear white as often
as the Comb was taken away; and then the Comb be¬
ing as was faid interpofed, that whitenefs by reafon of
the intercepted part of the Colours at the Lens did al¬
ways change into the Colour compounded of thofe
Colours which were not intercepted, and that Colour
was by the motion of the Comb perpetually varied fo,
that in the paffing of every Tooth over the Lens ail
thefe Colours red, yellow, green, blue and purple, did
always fucceed one another. I caufed therefore all the
Teeth to pafs fucceffively over the Lens, and when the
motion was How, there appeared a perpetual fuccefiion
of the Colours upon the Paper : But if I fo much acce¬
lerated the motion, that the Colours by reafon of their
quick fuccefiion could not be diftinguiftied from one
another, the appearance of the fingle Colours ceafed.
There was no red, no yellow, no green, no blue, nor
purple to be feen any longer, but from a confufion of
them all there arofe one uniform white Colour. Of the
Light which now by the mixture of all the Colours ap¬
peared white, there was no part really white. One
part was red, another yellow, a third green, a fourth
blue, a fifth purple, and every part retains its proper.
Colour till it ftnke the Senforium. If the imprefiions
follow one another (lowly, fo that they may be feve-
rally perceived, there is made a diftinft fenfation of all
the Colours one after another in a continual fuccefiion.

But...,

[io4]

But if the impreffions follow one another fo quickly
that they cannot be feverallv perceived, there arifeth
out of them all one common fenfation, which is nei¬
ther of this Colour alone nor of that alone, but hath it
felf indifferently to ’em all, and this is a fenfation of
whitenefs. By the quicknefs of the fucceffions the im¬
preffions of the feveral Colours are confounded in the
Senforium, and out of that confufion arifeth a mixt fen¬
fation, If a burning Coal be nimbly moved round in a
Circle with Gyrations continually repeated, the whole
Circle will appear like fire; the reafon of which is, that
the fenfation of the Coal in the feveral places of that
Circle remains impreft on the Senforium, until the
Coal return again to the fame place. And fo in a
quick confecution of the Colours the impreffion of every
Colour remains in the Senforium, until a revolution of
all the Colours be compleated, and that firft Colour re¬
turn again. The impreffions therefore of all the fucceffive
Colours are at once in the Senforium,and joyntly ftir up
a fenfation of them all; and fo it is manifeft by this Ex¬
periment, that the commixt impreffions of all the Co¬
lours do ftir up and beget a fenfation of white, that is,
that whitenefs is compounded of all the Colours,

And if the Comb be now taken away, that all the
Colours may at once pals from the Lens to the Paper,
and be there intermixed, and together reflected thence
to the Spectators Eyes ; their impreffions on the Senfo-
rium being now more lubtily and perfectly commixed
there, ought much more to ftir up a fenfation of white-
nefs*

You

[ 10 $] '

Y6u may inftead of the Lens ufe two Prifms HIK
and L M N, which by refracting the coloured Light
the contrary way to that of the firft refra&ion, may
make the diverging rays converge and meet again in G,
as you fee it repreiented in the feventh Figure. For Fig . y.
where they meet and mix they will compote a white
Light as when a Lens is ufed.

EXPEL XL

Let the Sun’s coloured Image PT fall upon the Wall Fig* 8*
of a dark Chamber, as in the third Experiment of the
firft Book, and let the fame be viewed through a Prifm
a be, held parallel to the Prifm ABC, by whofe refra¬
ction that Image was made, and let it now appear lower
than before, fuppofe in the place S over againft the red-
colour T. And if you go near to the Image PT, the
Speftrum S will appear oblong and coloured like the
Image P T • but if you recede from it, the Colours of
the Spectrum S will be contracted more and more, and
at length vaniih, that Spectrum S becoming perfectly
round and white ; and if you recede yet further, the
Colours will emerge again, but in a contrary order.

Now that SpeCtrum S appears white in that cafe when
the rays of leveral forts which converge from the feve-
ral parts of the Image PT, to the Prifm a be, are fo
retraCted unequally by it, that in their paffage from the
Prifm to the Eye they may diverge from one and the
lame point of the SpeCtrum S, and fo fall afterwards
upon one and the fame point in the bottom of the Eye,
and there be mingled.

And

Fig. 9.

[lod]

And further, if the Comb be here made ufe of, by
whole Teeth the Colours at the Image PT may be fuo
ceffively intercepted ; the Spedxum S when the Comb
is moved flowly will be perpetually tinged with fuc-
ceffive Colours: But when by accelerating the motion
of the Comb, the fucceffion of the Colours is fo quick
that they cannot be feverally feen, that Spedfrum S, by
a confufed and mixt lenfation of them all, will appear
white.

EXPER. XII.

The Sun Alining through a large Prifm ABC upon
a Comb X Y, placed immediately behind the Prifm, his
Light which paffed through the interftices of the Teeth
fell upon a whitePaper BE. The breadths of the
Teeth Were equal to their interftices, and feven Teeth
together with their interftices took up an Inch in
breadth. Now when the Paper was about two or
three Inches diftant from the Comb, the Light which
paffed through its feveral interftices painted fo many
ranges of Colours kl, mn, op, qr, tor. which were,
parallel to one another and contiguous, and without any
mixture of white. And thefe ranges of Colours, if the
Comb was moved continually up and down with a re¬
ciprocal motion, afcended and defcended in the Paper,
and when the motion of the Comb was fo quick, that
the Colours could not be diftinguilhed from one another,
the whole Paper by their confufion and mixture in the
Senforium appeared white*

Let

' C I0 7 3

Let the Comb now reft, and let the Paper be remo¬
ved further from the Prifin, and the feveral ranges of
Colours will be dilated and expanded into one another
more and more, and by mixing their Colours will di¬
lute one another, and at length, when the diftance
of the Paper from the Comb is about a Foot, or a
little more ( fuppofe in the place 2 D 2 E) they will
fo far dilute one another as to become white,,

With any Obftacle let all the Light be now ftopt
wdiich paffes through any one interval of the Teeth, lb
that the range of Colours which comes from thence may
be taken away, and you will fee the Light of the reft of
the ranges to be expanded into the place of the range
taken away, and there to be coloured. Let the inter¬
cepted range pafs on as before, and its Colours falling
upon the Colours of the other ranges, and mixing with
them, will reftore the whitenefs.

Let the Paper 2D 2E be now very much inclined to
the rays, fo that the mo ft refrangible rays may be more
copioufly reflected than the reft, and the white Colour
oi the Paper through the excefs of thofe rays will be
changed into blue and violet. Let the Paper be as
much inclined the contrary way, that the leaft refran¬
gible rays may be now more copioufly reflected than
the reft, and by their excefs the whitenefs will be
changed into yellow and red. The feveral rays there-
fore in that white Light do retain their colorific qua¬
lities, by which thole of any fort, when-ever they be-
come more copious than the reft, do by their excels
and predominance caufe their proper Colour to ap¬
pear, - x ;

And

[ io8 ]

And by the fame way of arguing, applied to the third
Experiment of this Book, it may be concluded, that
the white Colour of all refracted Light at its very firft
emergence, where it appears as white as before its inci¬
dence, is compounded of various Colours.

EX P ER. XIII.

In the foregoing Experiment the feveral intervals of
the Teeth of the Comb do the office of fo many Prifms.,
every interval producing the Phenomenon of one Prifm.
Whence inftead of thofe intervals ufing feveral Prifms, I
try'd to compound whitenefs by mixing their Colours,and
did it by ufing only three Prifms, as alfo by ufing only
fig. io. two as follows. Let two Prifms ABC and a be, whole
refrafting Angles B and b are equal,be fo placed parallel
to one another, that the refracting Angle B of the one
may touch the Angle c at the bafe of the other, and
their planes CB and cb, at which the rays emerge, may
lye in direftum. Then let the Light trajeCted through
them fall upon the Paper M M, diftant about 8 or 12
Inches from the Prifms. And the Colours generated
by the interior limits B and c of the two Prifms, will
be mingled at PT, and there compound white. For if
either Prifm be taken away, the Colours made by the
other will appear in that place PT, and when the Prifm
is reflated to its place again, fo that its Colours may
there fall upon the Colours of the other, the mixture
<*f them, both will reftore the whitenefs.

[ lop ]

This Experiment fueceeds alfo, as 1 have try ed, when
the Angle b of the lower Prifm, is a little greater than
the Angle B of the upper , and between the interior
Angles B an Ac, there intercedes fome fpace Be, as is
reprefented in the Figure, and the refracting planes
BC and be, are neither in dire&um, nor parallel to
one another. For there is nothing more requifite to
the fuccefs of this Experiment, than that the rays of all
forts may be uniformly mixed upon the Paper in the
place PT. If the mod refrangible rays coming from
the fuperior Prifm take up all the fpace from M toP, the
rays of the fame fort which come from the inferior
Prifm ought to begin at P, and take up all the reft of the
fpace from thence towards N. If the leaft refrangible
rays coming from the fuperior Prifm take up the fpace
M T, the rays of the fame kind which come from the
other Prifm ought to begin atT, and take up the remain¬
ing fpace T N». If one fort of the rays which have in¬
termediate degrees of refrangibility, and come from the
fuperior Prifm be extended through the fpace MQ, and
another fort of thofe rays through the fpace MR, and
a third fort of them through the fpace M S, the fame
forts of rays coming from the lower Prifm, ought to il¬
luminate the remaining fpaces QM, RN, SN refpe-
lively. And the fame is to be underftood of all the
other forts of rays. For thus the rays of every fort will
be fcattexed uniformly and evenly through the whole
fpace MN, and fo being every where mixt in the fame,
proportion, they muft every where produce the fame
Colour. And therefore fince by this mixture they pro-
duce white in the exterior fpaces M P and TN, fhey
muft alio produce white in the interior fpaceP T, This

[no]

is the reafon of the compofition by which whitenefs
was produced in this Experiment, and by what other
way foever I made the like compofition the refult was
whitenefs.

Laftly, If with the Teeth of a Comb of a due fize.
the coloured Lights of the two Prifms which fall upon
the fpace P T be alternately intercepted, that fpace
P T, when the motion of the Comb is flow, will always
appear coloured, but by accelerating the motion of
the Comb fo much, that the fucceflive Colours can¬
not be diftinguifiled from one another, it will appea r
white.

EXPEL XIV.

Hitherto I have produced whitenefs by mixing the
Colours of Prifms. If now the Colours of natural Bo¬
dies are to be mingled, let Water a little thickned with
Soap be agitated to raife a froth, and after that froth
has flood a little, there will appear to one that fliall
view it intently various Colours every where in the
furfaces of the feveral Bubbles; but to one that (hall
go fo far off that he cannot diftinguifh the Colours from
one another, the whole froth will grow white with a
perfect whitenefs.

EX PER. XV.

Laftly, in attempting to compound a white by mixing
the coloured Powders which Painters ufe, I confidered
that all coloured Powders do fupprefs and flop in
them a very confiderable part of the Light by which

they

[ill ]'

they are illuminated. For they become coloured by
reflecting the Light of their own Colours more copioufly,
and that of all other Colours more fparinglv, and yet
they do not refleCt the Light of their own Colours fo
copioufly as white Bodies do. If red Lead,for inftance,
and a white Paper, be placed in the red Light of the
coloured SpeCtrum made in a dark Chamber by the re¬
fraction of a Prifm, as is defcribed in the third Eperi-
ment of the firft Book ; the Paper will appear more lu¬
cid than the red Lead, and therefore reflects the red-
making rays more copioufly than red Lead doth. And
if they be held in the Light of any other Colour, the
Light reflected by the Paper will exceed the Light re¬
flected by the red Lead in a much greater proportion.
And the like happens in Powders of other Colours.
And therefore by mixing Inch Powders we are not to
expeCt a ftrong and full white, fuch as is that of Paper,
but fome dusky obfcure one, luch as might arife from a
mixture of light and darknefs, or from white and black,
that is, a grey, or dun, or ruffet brown, fuch as are the
Colours of a Man’s Nail, of a Moufe, of Allies, of or¬
dinary Stones, of Mortar, of Dull and Dirt in High¬
ways, and the like. And fuch a dark white I have
often produced by mixing coloured Powders. For thus
one part of red Lead,and five parts of Viride H£m,com-
pofed a dun Colour like that of a Moufe. For thefe
two Colours were feverally fo compounded of others,
that in both together were a mixture of all Colours ; and
there was lefs red Lead ufed than Viride ALrhj becaufe
of the fuinefs of its Colour. Again, one part of red
Lead, and four parts of blue Bife, compofed a dun Co¬
lour verging a little to purple, and by adding to this a

sextain

[ II2 1

certain mixture of Orpiment and ViricLi Arts in a due
proportion, the mixture loft its purple tinCture, and be¬
came perfectly dun. But the Experiment fucceeded beft
without Minium thus. To Orpiment I added by little
and little a certain full bright purple, which Painters
ule until the Orpiment ceafed to be yellow, and became
of a pale red. Then I diluted that red by adding a
little Vtride Arts, and a little more blue Bife than Vu
ridi Arm, until it became of luch a grey or pale white,
as verged to no one of the Colours more than to ano¬
ther. For thus it became of a Colour equal in white-
nefs to that of Allies or of Wood newly cut, or of a
Man's Skin. The Orpiment reflected more Light than
did any other of the Powders, and therefore conduced
more to the whitenefs of the compounded Colour than
they. To aflign the proportions accurately may be
difficult, by realon of the different goodnefs of Pow¬
ders of the fame kind. Accordingly as the Colour of
any Powder is more or lefs full and luminous, it ought
to be ufed in a lefs or greater proportion.

Now confidering that thefe grey and dun Colours
may be alfo produced by mixing whites and blacks, and
by confequence differ from perfeCt whites not in Species
of Colours but only in degree of luminoufnefs, it is ma-
nifeft that there is nothing more requifite to make
them perfeftly white than to increafe their Light fuffi-
ciently ; and, on the contrary, if by increafing their
Light they can be brought to perfeCt whitenefs, it will
thence alfo follow, that they are of the lame Species of
Colour with the beft whites, and differ from them only
in the quantity of Light. And this I tryed as follows.
I took the third of the above-mentioned grey mixtures

[i 13 3

( that Which was compounded of Qtpiment, Purple,
Bife and Vinde Mr is) and rubbed it thickly upon the
floor of my Chamber, where the Sun (hone upon it
through the opened Calement; and by it, in the (ha-
dow, I laid a piece of white Paper of the fame bignef^
Then going from them to the diftance of 1 2 or 18 Feet,"
fo that I could not difcern the unevennefs of the furface
of the Powder, nor the little (hadows let fall from the
gritty particles thereof; the Powder appeared intently
white, fo as to tranfcend even the Paper it felf in white-
nets, efpecially if the Paper were a little (haded from
the Light of the Clouds, and then the Paper compared
with the Powder appeared of fuch a grey Colour as the
Powder had done before. But by laying the Paper
where the Sun dimes through the Glafs of the Window,
or by (hutting the Window that the Sun might fhine
through the Glafs upon the Powder, and by fuch other
fit means of increafing or decreafing the Lights where¬
with the Powder and Paper were illuminated , the
Light wherewith the Powder is illuminated may be
made (Longer in fuch a due proportion than the Light
wherewith the Paper is illuminated, that they (hall both
appear exactly alike in whitenefs. For when I was
trying this, a Friend coming to vifit me, I ftopt him
at the door, and before I told him what the Colours
were, or what I was doing ; I askt him, Which of the
two whites were the beft, and wherein they differed ?
And after he had at that diftance viewed them well, he
anlwered, That they were both good whites, and that
he could not fay which was beft, nor wherein their Co¬
lours differed. Now if you confider, that this white
of the Ponder in the Sumfhine was compounded of the

P Colours

C r 14 1

Colours which the component Powders ( Orpiment r
Purple, Bile, and Viride JEris) have in the fame Sun-
fhine, you inuft acknowledge by this Experiment, as
well as by the former, that perfect whitenefs may be
compounded of Colours.

From what has been faid it is alfo evident, that thee
whitenefs of the Sun’s Light is compounded of all the
Colours wherewith the feveral forts of rays whereof
that Light confifts, when by their feveral refrangibili-
ties they are feparated from one another, do tinge Paper?
or any other white Body whereon .they fall. For thofer
Colours by Prop. 1. are unchangeable, and whenever
all thole rays with thofe their Colours are mixt again,
they reproduce, the fame white Light as before.

PROP, VI. PRGB. IL

M a mixture of primary Colour r, the quantity and quaiitfr
of, each -being given , ta know the Colour of the com-

With the Center O and Radius O B defcribe a Circle
A DF, and diftinguifti its circumference intodeven parts
DE, EE, FG$ GA, AB, BG, G D, proportional to
the feven mufical Tones or Intervals of the eight Sounds,
Sof la , /^, fol r la 1 fa , fof contained in an Eight,
that is, proportional to the numbers L, C

C Lev the fir ft part DE reprefent a red Colour, the
fecond E F orange, the third F G yellow,, the fourth
G H green, the fifth A B blue,. the fixth BC indico,
and the.feventh CD violet. And. conceive, that thefe

are all the Colours of uncompounded Light gradually

\ paffing

[”$]

paffing into one another, as they do when made by
Prifms ; the circumference DE FG A BCD, reprefen-
ting the whole l’eries of Colours from one end of the
Sun’s coloured Image to the other, fo that from D to E
be ail degrees of red, at E the mean Colour between red
and orange, from E to F all degrees of orange, at F the
mean between orange and yellow, from F to G all de¬
grees of yellow, and fo on. Let p be the center of
gravity of the Arch DE, and q, r, s, t, v, x, the centers
of gravity of the Arches EF, FG, GA, AB, BC
and C D refpeftively, and about thofe centers of gra¬
vity let Circles proportional to the number of rays of
each Colour in the given mixture be defcribed; that is,
the circle p proportional to the number of the red-ma¬
king rays in the mixture, the Circle q proportional to
the number of the orange-making rays in the mixture,
and fo of the reft. Find the common center of gravity
of all thofe Circles p, q, r, s, t, v, x. Let that center
be Z ; and from the center of the Circle A D F, through
Z to the circumference, drawing the right line O Y,
the place of the point Y in the circumference lhall fhew
the Colour arifing from the compofition of all the Co¬
lours in the given mixture, and the line O Z fnall be
proportional to the fulnefs or intenfenefs of the Colour,
that is, to its diftance from whitenefs. As if Y fall in
the middle between F and G, the compounded Colour
lhall be the beft yellow 3 if Y verge from the middle to¬
wards F or G, the compounded Colour lhall according¬
ly be a yellow, verging towards orange or green. IfZ
fall upon the circumference the Colour lhall be intenfe
and florid in thehigheft degree; if it fall in the mid
way between the circumference and center, it lhall be

P -2 but

[lid]

but half fo mtenfe, that is, it ihall be fucli a Colour as
would he made by diluting the intenfeft yellow with an
equal quantity of whitenefs; and if it fall upon the
center O, the Colour ihall have loft all its intenfenefiq
and become a white. But it is to be noted, That if the
point Z fall in or near the line O 33 , the main ingredients
being the red and violet, the Colour compounded ihall
not be any of the prifmatic Colours, but a purple, in¬
clining to red or violet, accordingly as the point Z
lieth on the fide of the line D O towards E or towards C,
and in general the compounded violet is more bright and
more fiery than the uncompounded. Alio if only two
of the primary Colours which in the Circle are oppofite
to one another be mixed in an equal proportion, the
point Z ihall fall upon the center O, and yet the Co¬
lour compounded of thofe two ihall not be perfectly
white, but fome faint anonymous Colour. For I could
never yet by mixing only two primary Colours produce
a perfect white. Whether' it may be compounded of a
mixture of three taken at equal diftances in the circum¬
ference I do not know, but of four or five I do not much
queftion but it may. But thefe are curiofities of little
or no moment to the underftanding the Phenomena of
nature. For in all whites produced by nature, there
ufes to be a mixture of all forts of rays, and by confe-
quence a competition of all Colours.

To give an inftance of this Rule; fuppofe a Colour is
compounded of thefe homogeneal Colours, of violet
i part, of indico i part, of blue a parts, of green 3 parts,
of yellow 5 parts, of orange 6 parts, and of red 1 o parts.
Proportional to thefe parts I deferibe the Circles x, v, t,
% r i T P refpedtively, that is, la that if the Circle x

[ ri 7 ] _

be i, the Circle v may be i, the Circle t r, the Circle
S3, and the Circles r, q and p, 5, 6 and 10. Then I
find Z the common center of gravity of thefe Circles,
and through Z drawing the line OY, the point Y falls
upon the circumference between E and F, fome thing
nearer to E than to F, and thence I conclude, that the
Colour compounded of thefe ingredients will be an
orange, verging a little more to red than to yellow.
Alfo I find that O Z is a little lefs than one half of
OY, and thence 1 conclude, that this orange hath a
little lels than half the fulnefs or intenfenefs of an un¬
compounded orange ; that is to fay, that it is fuch an
orange as may be made by mixing an homogeneal orange-
with a good white in the proportion of the line O Z to
the line Z Y, this proportion being not of the quantities
of mixed orange and white powders, but of the quan¬
tities of the lights refiedted from them.

This Rule 1 conceive accurate enough for pradtife,
though not mathematically accurate ; and the truth of
it may be fufficiently proved to fenfe, by flopping any
of the Colours at the Lens in the tenth Experiment of
this Book. For the reft of the Colours which are not
flopped, but pafs on to the Focus of the Lens, will
there compound either accurately or very nearly fuch
a Colour as by this Rule ought to refult from them
mixture.

PRO Vi

.• » —I

PROP. VII. THE OR* V.

>,+411 the Colours in the Univerje -which are made by Light ,
and depend not on the power of imagination , are
either the Colours of homogeneal Light r, or compounded
of thefe and that either accurately or very nearly , ac¬
cording to the Rjule of the foregoing Tdr&blem.

For it has been proved ( in Prop.i. Lib.o..) that the
^changes of dolours made by refradions do not arife
from any new modifications of the rays impreft by thofe
refradions, and by the various terminations of light
andfhadow, as has been the conftant and general opi¬
nion of Philofophers. It has alfo been proved that the
leveral Colours of the homogeneal rays do conftantly
anfwer to their degrees of refrangibility, (Prop, i. Lik i.
and Prop.i. Lib.'i.) and that their degrees of refrangi-
bility cannot be changed by refradions and reflexions,
(Prop.i. Lib A.) and by confequence that thofe their
Colours are likewife immutable. It has alfo been pro¬
ved diredly by refrading and refieding homogeneal
Tights apart, that their Colours cannot be changed,
(Prop.i. Lib.o..) It has been proved alfo, that when
the feveral forts of rays are mixed, and in croffing ; pafs
through the fame fpace, they do not ad on one another
So as to change each others colorifick qualities, (Exper.
lo.Lik'i.) but by mixing their adions in the Senfio-
rium beget a ienfation differing from what either would
do apart, that is a fenfation of a mean Colour between
fheir proper Colours ; and particularly when by the
^oncourfe and .mixtures of all forts of rays, a white

Colour

[H5>]

Colour is produced, the white is a mixture of all the
Colours which the rays would have apart, ( Prop. 5,
Lik c. ) The rays in that mixture do not lofe or alter
their feveral colorifick qualities, but by all their various
kinds of actions mixt in the Senforium r beget a fenfa-
tion of a middling Colour between all their Colours-
which is whitenefs. For whitenefs is a mean between ■
all Colours, having it felf indifferently to them aFg fo-
as with equal facility to be tinged with any of them.
A red Powder mixed with a little blue, or a blue with
a little red, doth not prelently lofe its Colour, but a
white Powder mixed with any Colour is prefently tin¬
ged with that Colour, and is equally capable of being
tinged with any Colour what-ever. It has been (hewed ’
aifo, that as the Sun’s Light is mixed of all forts of rays,
lb its whitenefs is a mixture of the Colours of all forts.*-
of rays; thofe rays having from the beginning their fe-
veral colorific qualities as well as their feveral refrangL
bilities, and retaining them perpetually unchang’d not-
withftanding any refractions or reliexions they may at
any time fuffer, and that whemever any fort of the
Sun’s-rays is by. any means (as by reflexion in Ex per. y,
audio. Lik 1. or by refraCtion as happens in all re¬
fractions) feparated from the : reft, they then manifeft
their proper Colours, ... Thefe things have been proved/
and the fum of all this amounts to the Propofition here
to be proved.. For if the Sun’s Light is mixed of 1L, ~
veral forts of rays,- each of which have originally their
feveral refrangibilities and colorifick qualities, and not- -
withftanding their refraCtions and reflections, and their .
various reparations or . mixtures, keep .thofe their ori- -
ginal properties perpetually the fame without, altera¬
tion ] ;

[ 120 ]

"tion ; then all the Colours in the World mu ft: be fuch as
conftantly ought to arife from the original colorific qua¬
lities of the rays whereof the Lights coniift by which
thofe Colours are feen. And therefore if the reafon of
any Colour what-ever be required., we have nothing elfe
to do then to confider how the rays in the Sun’s Light
have by reflexions or refractions, or other caufes been par¬
ted from one another,or mixed together; or otherwile to
find out what forts of rays are in the Light by which
that Colour is made, and in what proportion; and
then by the laft Problem to learn the Colour which
ought to arife by mixing thofe rays (or their Colours)
in that proportion. I fpeak here of Colours fo Aar as
they arife from Light. For they appear fometimes by
other caufes, as when by the power of phantafy we
fee Colours in a Dream, or a mad Man fees things before
him which are not there ; or when we fee Fire by ftriking
the Eye, or fee Colours like the Eye of a Peacock’s
Feather, by preffing our Eyes in either corner whilft
we look the other way. Where thefe and fuch like
caufes interpofe not, the Colour always anfwers to
the fort or forts of the rays whereof the Light confiffs,
as I have conftantly found in what-ever Phenomena of
Colours I have hitherto been able to exarniir 1 flrall in
the following Propoiitions give infiances of this in the
Phenomena of ehiefeft note.

PROP.

PROP. VIII. P ROB. III.

By the dijcovered ‘Properties of Light to explain the

Colours made by Prifms .

Let ABC reprefent a Prifm refracting the Light of pi a
the Sun, which comes into a dark Chamber through a °
Hole F <P almoft as broad as the Prifm, and let M N
reprefent a white Paper on which the refracted Light is
caft, and fuppofe the moft refrangible or deepeft violet
making rays fall upon the fpace Ptt , the lea ft refran¬
gible or deepeft red-making rays upon the fpace T
the middle fort between the Indico-making aud blue¬
making rays upon the fpace Q x , the middle fort of the
green-making rays upon the fpace R e, the middle fort
between the yellow-making and orange-making rays
upon the fpace S and other intermediate forts upon
intermediate fpaces. For fo the fpaces upon which the
feveral forts adequately fall will by reafon of the diffe¬
rent refrangibility of thofe forts be one lower than ano¬
ther. Now if the Paper MN be fo near the Prifm that the
fpaces P T and ■rri do not interfere with one another, the
diftance between them T T will be illuminated by all
the l'orts of rays in that proportion to one another which
they have at their very firft coining out of the Prifm,
and consequently be white. But the fpaces PT and
on either hand, will not be illuminated by them all,
-and therefore will appear coloured. And particularly
at P,^ where the outmoft violet-making rays fall alone,
the Colour muft be the deepeft violet. At Q where the
violet-making and indico-making rays are mixed, it

Q, muft

„ p 22 3

muft be a violet inclining much to indico. At R where
the violet-making, indico-making, blue-making, and
one half of the green-making rays are mixed, their Co¬
lours muft ( by the conftruftion of the fecund Problem)
compound a middle Colour between indico and blue.
At S where all the rays are mixed except the red-ma¬
king and orange-making,their Colours ought by the fame
Rule to compound a faint blue, verging more to green
than indie. And in the progrefs from S to T, this blue
will grow more and more faint and dilute, till atT,
where all the Colours begin to be mixed , it end in
whitenefs.

So again, on the other fide of the white at T, where
the leaft refrangible or utmoft red-making r ays are alone
the Colour muft be the deepeft red. At cr the mixture
of red and orange will compound a red inclining to
orange. At e the mixture of red, orange, yellow, and
one half of the green muft compound a middle Colour
between orange and yellow. At x the mixture of all
Colours but violet and indico will compound a feint
yellow, werging more to green than to orange. And
this yellow will grow more feint and dilute continually
in its progrefs from ^ to tt, where by a mixture of all
forts of rays it will become white.

Thefe Colours ought to appear were the Sun's Light
perfectly white: But becauleit inclines to yellow,the ex¬
cels of the yellow-making rays whereby ftis tinged with
that Colour, being mixed with the faint blue between
S and T, will draw it to. a faint green. And fo the
Colours in order from P toT ought to be violet, indico,
blue, very feint green, white, feint yellow, orange, red.
Thus it is by the computation : And they that pleafe to

view

[ I2 3 J

view the Colours made by a Prifm will find it fo in
Nature.

Thele are the Colours on both fides the white when
the Paper is held between the Prifm, and the point X
where the Colours meet, and the interjacent white va-
nifhes. For if the Paper be held ftill farther off from the
Priirn, the molt refrangible and leaft refrangible rays
will be wanting in the middle of the Light, and the reft
of the rays which are found there, will by mixture pro¬
duce a fuller green than before. Alfo the yellow and
blue will now become lefs compounded, and by con-
fequence more intenfe than before. And this alio
agrees with experience.

And if one look through a Prifm upon a white Object
encompaffed with blacknefs or darknefs, the reafon of
the Colours arifing on the edges is much the fame, as
will appear to one that fhall a little corifider it. If a
black Object be encompaffed with a white one, the Co¬
lours which appear through the Prifm are to be derived
from the Light of the white one, fpreading into the Re¬
gions of the black, and therefore they appear in a con¬
trary order to that, in which they appear when a white
Object is furrounded with black. And the fame is to
be underftood when an Objeft is viewed, whofe parts
are l'ome of them lefs luminous than others. For in the
Borders of the more and lefs luminous parts, Colours
ought always by the fame Principles to arife from the
excels of the Light of the more luminous, and to be of
the fame kind as if the darker parts were black, but yet
to be more faint and dilute.

C !24]

What is faid of Colours made by Prifms may be eafily
applied to Colours made by the Glafifes of Telefeopes,
or Microfcopes, or by the humours of the Eye. For if
the ObjeCt-glafs of a Telefcope be thicker on one fide
than on the other, or if one half of the Glafs, or one
half of the Pupil of the Eye be covered with any opake
fubftance: the ObjeCt-glafs, or that part of it or of the
Eye which is not covered, may be confidered as a Wedge
with crooked fides, and every Wedge of Glafs, or other
pellucid fubftance, has the effeCt of a Prifm in refracting
the Light which paflfes through it.

How the Colours in the 9th and 10th Experiments
of the firft Part arile from the different reflexibility of
Light,is evident by what was there faid. But it is obfer-
vable in the 9th Experiment, that whilft the Sun's di-
reCt Light is yellow, the excefs of the blue-making
rays in the reflected Beam of Light MM, fuffices only

to bring that yellow to a pale white inclining to blue,
and not to tinge it with a manifeftly blue Colour. To -
obtain therefore a better blue, I ufed inftead of the yel¬
low Light of the Sun the white Light of the Clouds, by
varying a little the Experiment as follows*

EX PER. XVI,

Let H F G reprefent a Prifm in the open Air, and S
the Eye of the Spectator, viewing the Clouds by their
Light coming into the Prifm at the plane fide BIGK,
and reflected in it by its bale H EIG, and thence going
out through its plain fide H E F K to the Eye. And
when the Prifm and Eye are conveniently placed, fo
that the Angles of incidence and reflexion at the bafe

[* 2 $]

may be about 40 degrees, the Spectator will fee a Bow
M N of a blue Colour, running from one end of the
bale to the other, with the concave fide towards him,
and the part of the bafe IJMN G beyond this Bow will
he brighter than the other part EMNH on the other
fide or it. This blue Colour MN being made by no¬
thing elfe than by reflexion of a fpecular fuperficies,
feems fo odd a Phenomenon, and fo unaccountable for
by the vulgar Hypothefis of Philofopliers, that 1 could
not but think it deferved to be taken notice of . Now
for underftanding the reafon of it, fuppofe the plane
ABC to cut the plane {ides and bafe of the Prifm per¬
pendicularly. From the Eye to the line BC, wherein that
plane cuts the bafe, draw the lines Sp and S t, in the ?
Angles Spc 50 degr. and Stc 49 degr,- z8 , and the
point f will be the limit beyond which none of the moft
refrangible rays can pafs through the bafe of the Prifm,
and be refracted, whole incidence is fuch that they may
be reflected to the Eye; and the point t will be the like
limit for the leaft refrangible rays, that is, beyond ;
which none of them can pafs through the bafe, whofe -
incidence is fuch that by reflexion they may come to the
Eye. And the point r taken in the middle way between
p and t, will be the like limit for the meanly refrangible
rays. And therefore all the refrangible rays which fall
upon the bafe beyond t, that is, between t and B, and
can come from thence to the Eye will be reflected thi¬
ther : But on this fide t, that is, between t and c, many „
of thefe rays will be tranfmitted through the bafe.
And all the moft refrangible rays which fall upon the
Bale beyond p, that is, between p andB, and can by
reflexion come from thence to the Eye, will be reflected

thither^ %

[ I2J5~]

thither, but every where between t and c, many of
thefe rays will get through the bafe and be refracted ;
and the fame is to be underftood of the meanly refran¬
gible rays on either fide of the point r. Whence it fol¬
lows, that the bafe of the Prifin muft every where be¬
tween t and B, by a total reflexion of all forts of rays to
the Eye, look white and bright And every where
between p and C, by reafon of the tranfmiflion of many
rays of every fort, look more pale, obfcure and dark.
But at r, and in other places between p and t, where
all the more refrangible rays are reflefted to the Eye,
and many of the lefs refrangible are tranfmitted, the
excefs of the mod refrangible in the reflected Light will
tinge that Light with their Colour, which is violet and
blue* And this happens by taking the line CprtB any
where between the ends of the Prifin H G and E L

PROP. IX. P ROB. IV.

J 3 y the dijcovered ‘Properties of Light to explain the

Colours of the Rjim^iow,

This Bow never appears but where it Rains in the
iSun-fhine, and may be made artificially by fpouting up
Water which may break aloft, and lcatter into Drops,
and fall down like Rain. For the Sun Ihining upon thefe
Drops certainly caufes the Bow to appear to a Specta¬
tor handing in a due pofition to the Rain and Sun. And
hence it is now r agreed upon, that this Bow is made by
refraCfion of the Sun’s Light in Drops of falling Rain.
This was underitood by fome of the Ancients, and of

the Famous
Ant on in s

late more fully difcovered and explained by

't127 J

Antonius de Donums Archbifhop of Splaics. in his Book
T)e Radiis V'tfm Lticis , publifhed by his Friend

tolm at JS'enice^ in the Year 1611 ^ and written above
twenty Years before. For he teaches there how the
interior Bow is made in round Drops of Rain by two
refractions of the Sun's Light, and one reflexion be¬
tween them, and the exterior by two refractions and
two forts of reflexions between them in each Drop of
Water, and proves his Explications by Experiments
made with a Phial full of Water,and with Globes ofGlafs
filled with Water, and placed in the Sun to make the
Colours of the two Bows appear in them. The fame
Explication Bes^Cartes hath purfued in his Meteors,
and mended that of the exterior Bow. But whilft they
underftood not the true origin of Colours, it’s neceffary
to purfue it here a little further. For underftanding
therefore how the Bow is made, let a Drop of Rain or
any other fpherical tranfparent Body be reprefented by
the Sphere B N F G, defcribed wflth the Center C, and Rg*.\W
Semi-diameter GN. And let AN be one of the Sun’s
rays incident upon it at N, and thence refraCted to F,
where let it either go out of the Sphere by refraCtion to¬
wards V, or be reflected to G ; and at G let it either go
out by refraCtion to R, or be reflected to H ; and at H
let it go out by refraCtion towards S, cutting the inci¬
dent ray in Y;; produce AN and RG, till they meet in
X, and upon AX and NF let fall the perpendiculars
CD and CE, and produce CD till it fall upon the cir-
eumlerence at L. Parallel to the incident ray AN draw?,
the Diameter B Q, and let the fine of Incidence out of
Air into Water be to the fine of refraCtion as I to,

Ro Now if you fuppofe the point of incidence N to

move...

[ 128]

move from the point B, continually til! It come to T,
the Arch QF will firft increafe and then decreafe, and
fo will the Angle A X R which the rays A N and G R

- contain; and the Arch Q F and Angle A X R will be
biggeft when ND is to CN as KnTRR to 7/3 RR,

- in which cafe N E will be to N D as 2 R to I. Alfo the
Angle AYS which the rays A N and H S contain will
lirft decreafe, and then increafe and grow leaft when
N D is to C N as // iRrr to R R, in which cafe
M E will be to ND as 3 R toI. And fo the Angle which

■ the next emergent ray (that is, the emergent ray after
three reflexions ) contains with the incident ray A N
will come to its limit whenND is to CN as / ii-rr to
/A 5 RR, in which cafe N E will be to ND as 4R to I,
and the Angle which the ray next after that emergent,
that is, the ray emergent after four reflexions, con¬
tains with the incident will come to its limit, when
ND is to CN/ as / iErr to /A4 RR, in which cafe
JME will be to N D as 5 R to I; and fo on infinitely^
the numbers 3, 8,15, 14, &c. being gathered by conti¬
nual addition of the terms of the arithmetical progreffion
3,5,7,9,^. The truth of all this Mathematicians will
ceafily examine.

Now it is to be obferved, that as when the Sun comes
to his Tropicks, days increafe and decreafe but a very
little for a great while together ; fo when by increasing
; the diftance C D, thefe Angles come to their limits,
they vary their quantity but very little for fome time
together, and therefore a far greater number of the rays
which fall upon all the points N in the Quadrant
IBL, (hall emerge in the limits of thefe Angles,
..then in any other inclinations. And further it is

[ 129 ]

to be obferved, that the rays which differ in refrangi-
bility will have different limits of their Angles of emer¬
gence, and by confequence according to their different
degrees of refrangibility emerge mod copioufly in dif¬
ferent Angles, and being feparated from one another
appear each in their proper Colours. And what thole
Angles are may be eafily gathered from the foregoing
Theorem by computation.

For in the leaft refrangible rays the fines I and R (as
was found above) are 108 and 81, and thence by
computation the greateft Angle AXR will be found
4a degrees and a minutes, and the leaft Angle AYS,
go degr. and 57 minutes. And in the moft refrangible
rays the fines I and R are 109 and 81, and thence by
computation the greateft Angle AXR will be found
4.0 degrees and 17 minutes, and the leaft Angle AYS
54 degrees and 7 minutes.

Suppofe now that O is the Spedtator’s Eye, and OP a line fig
drawn parallel to the Sun's rays, and let P O E, P O F,
POG, POH, be Angles of 40 degr. 17 min. 4a degr.

2 min. 50 degr. 57 min. and 54 degr. 7 min. refpedtively,
and thefe Angles turned about their common fide O P,
ftiall with their other fidesOE, OF; OG, OH de-
defcribe the verges of two Rain-bows A F BE and
CHDG. For if E, F, G, H, be Drops placed any
where in the conical fuperficies defcribed by OE, OF,

O G, O H, and be illuminated by the Sun’s rays S E,

SF, SG, SH; the Angle SEO being equal to the
Angie POE or 40 degr. 17mm. ftiall be the greateft
Angle in which the moft refrangible rays can after one
reflexion be refradted to the Eye, and t herefore all the
Drops in the line O E {hall fend the moft refrangible

R rays

C r 3° ]

rays mo ft copioufly to the Eye, and thereby ftrike the
fenles with the deepeft violet Colour in that region.
And in like manner the Angle SFO being equal to*
the Angle P O F, or 42 deg. 2 min, (hall be the greateft
in which the leaft refrangible rays after one reflexion
can emerge out of the Drops., and therefore thofe rays
fhafl come molt copioufly to the Eye from the Drops in
the line O F, and ftrike the fenfes with the deepeft red
Colour in that region. And by the fame armament,
the rays which have intermediate degrees of refrangibi-
lity fhall come molt copioufly from Drops between
E and F, and ftrike the fenfes with the intermediate
Colours in the order which their degrees of refrangibi-
lity require, that is, in the progrefs from E to F, or
from the infide of the Bow to the outiide in this order,
violet, indico, blue, green, yellow,orange, red. But the
violet, by the mixture of the white Light of the Clouds,
will appear faint and incline to purple.

Again, the Angle SCO being equal to Angle P O G,
or 50 gr. 51 min. fhall be the leaft Angle in which the
leaft refrangible rays can after two reflexions emerge out
of the Drops,and therefore the leaft refrangible rays fhall
come moft copioufly to the Eye from the Drops in the
lineOG, and ftrike the feme with the deepeft red in
that region. And the Angle S H O being equal to the
Angle P O H or 54 gr. 7 min. fhall be the leaft Angle in
which the moft refrangible rays after two reflections can
emerge out of the Drops, and therefore thofe rays fhall
come moft copioufly to the Eye from the Drops in the
line O H, and ftrike the fenfes with the deepeft violet in
that region. And by the fame argument, the Drops in
the regions between G and H fhall ftrike the fenfe with

the

[igi]

the intermediate Colours in the order which their de¬
grees of refrangibility require, that is, in the progress
from G to H, or from the infide of the Bow to the out-
fide in this order, red, orange, yellow, green, blue, in-
dico, violet. And fince thefe four lines O E, O F, O G.
O H, may be fituated any where in the above-mentioned
conical fuperficies, what is faid of the Drops and Co¬
lours in thefe lines is to be underftood of the Drops
and Colours every where in thofe fuperficies.

Thus fhall there be made two Bows of Colours, an
interior and ftronger, by one reflexion in the Drops,
and an exterior and fainter by two • for the Light be¬
comes fainter by every reflexion. And their Colours
fhall ly in a contrary order to one another, the red of
both Bows bordering upon the fpace G F which is be¬
tween the Bows. The breadth of the interior Bow
EOF meafured crofs the Colours fhall be I degr. 45 min.
and the breadth of the exterior GOH fhall be 3
degr. 1 o min. and the diftance between them GO F
fhall be 8 gr. 5 5 min. the greateft Semi-diameter of the
innermoft, that is, the Angle POF being 42 gr. a min.
and the lea ft Semi-diameter of the outermoft P O G, be¬
ing 50 gr. 57 min. Thefe are the meafures of the Bows,
as they would be were the Sun but a point; for by the
breadth of his Body the breadth of the Bows will be in-
creafed and their diftance decreafed by half a degree,
and fo the breadth of the interior Iris will be 2 degr.

* S min. that of the exterior 3 degr. 40 min. their dj~
ftance 8 degr. 35 min. the greateft Semi-diameter of the
interior Bow 42 degr. 17 min. and the leaft of the ex¬
terior 50 degr. 43 min. And fuch are the diraenfions
of the Bows in the Heavens found to be verv nearly,

2 when

[wy]

when their Colours appear ftrong and perfect. For
once, by Inch means as I then had, I meafured the
greateft Semi-diameter of the interior Iris about 4a de¬
grees, the breadth of the red, yellow and green m that
Iris 63 or 64 minutes, betides the outmoft faint red ob-
fcured by brightnefs of the Clouds, for which we
may allow 3 or 4 minutes more. The breadth of the
blue was about 40 minutes more betides the violet,
which was fo much obfcured by the brightnefs of the
Clouds, that I could not meafure its breadth. But
fuppofing the breadth of the blue and violet together
to equal that of the red, yellow and green together, the
whole breadth of this Iris will be about a- degrees as
above. The leaft diftance between this Iris and the ex¬
terior Iris was about 8 degrees and 30 minutes. The ex¬
terior Iris was broader than the interior, but fo faint,
efpeciaily on the blue fide, that I could not meafure its
breadth diftindtly. At another time when both Bows
appeared more diftindt, I meafured the breadth of the
interior Iris 2 gr. 1 04 and the breadth of the red, yel¬
low and green in the exterior Iris, was to the breadth
of the fame Colours in the interior as 3 to a.

This Explication of the Rain-bow is yet further con¬
firmed by the known Experiment ( made by Antonins
de Domints and ‘Des-Cartes) of hanging up any where
in the Sun-fhine a G 1 ifs-Globe filled with Water, and
viewing it in fuch a pofture that the rays which come
from the Globe to the Eye may contain with the Sun’s
rays an Angle of either 4a or 50 degrees. For if the
Angle be about 4a or 43 degrees, the Spectator ( fup-
pofe at O) (hall lee a full red Colour in that fide of the
Globe oppofed to the Sun as ’tis reprefented at F, and

[133]

if that Angle become lefs ( fuppofe by depreffing the
Globe to E ) there will appear other Colours, yellow,
green and blue fucceffively in the fame fide of the Globe.
But if the Angle be made about 50 degrees (fuppofe by
lifting up the Globe to G)there will appear a red Colour
in that fide of the Globe towards the Sun, and if the
Angle be made greater (fuppofe by lifting up the Globe
to H) the red will turn fucceffively to the other Colours
yellow, green and blue. The fame thing I have tried by
letting a Globe reft, and railing or depreffing the Eye,,
or otherwife moving it to make the Angle of a Juft
magnitude.

I have heard it reprefen ted, that if the Light of a
Candle be refracted by a Prifm to the Eye ; when the
blue Colour falls upon the Eye the Spectator ftiall fee
red in the Prifm, and when the red fills upon the Eye
he ftiall fee blue ; and if this were certain, the Colours
of the Globe and Rain-bow ought to appear in a con¬
trary order to what we find. But the Colours of the
Candle being very faint, the miftake feems to arife from
the difficulty of difcerning what Colours fall on the
Eye. For, on the contrary, I have fometimes had oc~
calion to obferve in the Sun’s Light refradted by a Prifm,
that the Spectator always fees that Colour in the Prifm
which falls upon his Eye. And the fame I have found
true alfo in Candle-Light. For when the Prifm is mo¬
ved ilowly from the line which is drawn diredtly from the
Candle to the Eye, the red appears fir ft in the Prifm and
then the blue, and therefore each of them is feen when
it falls upon the Eye. For the red paffes over the Eye
firft, and then the blue.

The

t 1 34 3

The Light which comes through Drops of Rain by
two refractions without any reflexion, ought to appear
ftrongeft at the diftance of about a 6 degrees from the
Sun, and to decay gradually both ways as the diftance
from him increafes and decreafes. And the fame is to
be underftood of Light tranfmitted through fphericai
Hail-ftones. And if the Hail be a little flatted, as it
often is, the Light tranfmitted may grow fo ftrong at
a little left diftance than that of a6 degrees, as to form
a Halo about the Sun or Moon ; which Halo, as often
as the Hail-ftones are duly figured may be coloured,,
and then it muft be red within by the leaft refrangible
rays,and blue without by the moft refrangible ones,efpw
dally if the Hail-ftones have opake Globules of Snow in
their center to intercept the Light within the Halo ( as
Trlugemm has obferved) and make the infid e thereof more
diftinCbly defined than it would otherwife be. For
fuch Hail-ftones, though fphericai, by terminating the
Light by the Snow, may make a Halo red within and
colourlefs without, and darker in the red than with¬
out, as Halos ufe to be. For of thole rays which pa ft
clofe by the Snow the rubriform will be leaft refraCted,
and fo come to the Eye in the diredeft lines.

The Light which paffes through a Drop of rain after
two refradions, and three or more reflexions, is icarce
ftrong enough to caule a fen Able Bow j but in* thole Cy¬
linders of Ice by which Hugenim explains the Tewbelm
it may perhaps be fen fible.

PROP

» •"

C 1 3 5II

PROP. X. PROS. V.

Bi the dijcovered'Properties $f Light to explain the per*
raanent Colours of natural Bodies.

Theie Colours arife from hence, that fome natural
Bodies reflect fome forts of rays, others other forts more
copioufly than the reft. Minium reflects the leaft re¬
frangible or red-making rays moft copioufty, and thence
appears red. Violets reflect the moft refrangible, moft
copioufly, and thence have their Colour, and fo of other
Bodies. Every Body reflects the rays of its own Colour
more copioufly than the reft, and from their excefs and
predominance in the reflected Light has its Colour.

EX PER. XV XL

For if the homogeneal Lights obtained by the folia¬
tion of the Problem propofed in the ^.th Propofition of
the firft Book you place Bodies of feveral Colours, you
will find, as X have done, that every Body looks moft
iplendid and luminous in the Light of its own Colour.
Cinnaber in the homogeneal red Light is moft refplen-
den-t, in the green Light it is manifeftly lefs refplen-
dent, and in the blue Light ftill left. Ind-ico in the
violet blue Light is moft refplendent, and its fplendor
is gradually diminiftied as it is removed thence by de¬
grees through the green and yellow Light to the red.

green Light, and next that the blue and,
compound green, are more ftrongly re¬
flected

By a Leek the
yellow which

flefted than the other Colours red and violet,and fo of the
reft. But to make thefe Experiments the more manifeft,
fuch Bodies ought to be chofen as have the fulleft and
moil vivid Colours, and two of thole Bodies are to be
compared together. Thus, for inftance, if Cinnaber
and ultra marine blue, or fome other full blue be
held together in the homogeneal Light, they will both
appear red, but the Cinnaber will appear of a ftrongly
luminous and refplendent red, and the ultra marine
blue of a faint obfcure and dark red; and if they be
held together in the blue homogeneal Light they will
both appear blue, but the ultra marine will appear of
a ftrongly luminous and refplendent blue, and the
Cinnaber of a faint and dark blue. Which puts it out
of difpute, that the Cinnaber reflects the red Light
much more copioully than the ultra marine doth, and
the ultra marine refledts the blue Light much more co-
pioufly than the Cinnaber doth. The fame Experiment
may be tryed fuccesfully with red Lead and Indico, or
with any other two coloured Bodies, if due allowance
be made for the different ftrength or weaknefs of their
Colour and Light.

And as the reafon of the Colours of natural Bodies is
evident by thefe Experiments, fo it is further confirmed
and put paft difpute by the two firft Experiments of the
firftBook, whereby ’twas proved in fuch Bodies that
the reflected Light which differ in Colours do differ alfo
in degrees of refrangibility. For thence it’s certain,
that fome Bodies refieft the more refrangible, others
the lefs refrangible rays more copioufly.

And

0371

And that this is not only a true reafon of thefe Co¬
lours, but even the only reafon may appear further
from this conlideration, that the Colour of homogeneal
Light cannot be changed by the reflexion of natural

Bodies.

For if Bodies by reflexion cannot in the leaf! change
. the Colour of any one fort of rays, they cannot appear
coloured by any other means than by reflecting thofe
which either are of their own Colour, or which by
mixture muft produce it.

But in trying Experiments of this kind care muft be
had that the Light be fufficiently homogeneal. For if
Bodies be illuminated by the ordinary prifmatick Co¬
lours, they will appear neither of their own day-light
Colours, nor of the Colour of the Light caft on them,
but of lbme middle Colour between both, as I have
found by Experience. Thus red Lead ( for inftance )
illuminated with the ordinary prifmatick green will
not appear either red or green, but orange or yellow,
or between yellow and green accordingly, as the green
Light by which ’tis illuminated is more or lefs com¬
pounded. For becaufe red Lead appears red when il¬
luminated with white Light, wherein all forts of rays
are equally mixed, and in the green Light all forts of
rays are not equally mixed, the excels of the yellow¬
making, green-making and blue-making rays in the
• incident green Light, will caufe thofe rays to abound
to much in the reflected Light as to draw the Colour
from red towards their Colour. And becaufe the red
Lead reflects the red-making rays moft copioufly in
proportion to their number, and next after them the
orange-making and yellow-making rays • thefe rays in

S the

[ x 3 8 J

the refiefted Light will be more in proportion to the
Light than they were in the incident green Light, and
thereby will draw the reflected Light from green to¬
wards their Colour. And therefore the red Lead will ap¬
pear neither red nor green,butofaColour between both.

In tranfparently coloured Liquors ’tis obfervable,
that their Colour ufes to vary with their thicknefs.
Thus, for inftance, a red Liquor in a conical Glafs
held between the Light and the Eye, looks of a pale
and dilute yellow at the bottom where ’tis thin, and a
little higher where ’tis thicker grows orange,and where
’tis ftill thicker becomes red, and where ’tis thickeft
the red is deepeft and darkeft. For it is to be conceived
that fuch a Liquor flops the indico-making and violet-
making rays moft eafily, the blue-making rays more
difficultly, the green-making rays ftill more difficultly,
and the red-making moft difficultly : And that if the
thicknefs of the Liquor be only fo much as fuffices to
flop a competent number of the violet-making and in¬
dico-making rays, without diminifhing much the num¬
ber of the reft, the reft muft (by Prop. 6. Lib. a.) com¬
pound a pale yellow. But if the Liquor be fo much
thicker as to flop alfo a great number of the blue-making
rays, and fome of the green-making, the reft muft com¬
pound an orange ; and vrhere it is fo thick as to flop
alfo a great number of the green-making and a consi¬
derable number of the yellow-making, the reft muft
begin to compound a red, and this red muft grow deeper
and darker as the yellow making and orange-making
rays are more and more ftopt by increasing the thick¬
nefs of the Liquor, fo that few rays befides the red-
making can get through,,

[ 1 39 ]

Of this kind is an Experiment lately related to me by
Mr. Halle y, who, in diving deep into the Sea, found
in a clear Sundhine day, that when he was funk many
Fathoms deep into the Water, the upper part of his
Hand in which the Sun flione diredly through the
Water looked of a red Colour, and the under part of
his Hand illuminated by Light reflected from the Water
below looked green. For thence it may be gathered,
that the Seacwater reflects back the violet and blue¬
making rays moft eafily, and lets the red-making rays
pafs moft freely and copiouily to great depths. For
thereby the Sun’s direCt Light at all great depths, by
reafon of the predominating red-making rays, muft
appear red ; and the greater the depth is, the fuller
and intenfer muft that red be. And at fuch depths as
the violet-making rays fcarce penetrate unto, the blue-
making, green-making and yellow-making rays being
reflected from below niore copiouily than the red~making
ones, muft compound a green.

Now if there be two Liquors of full Colours, fup-
pole a red and a blue, and both of them fo thick as
fuffices to make their Colours fufficiently full ; though
either Liquor be fufficiently tranfparent apart, yet
will you not be able to fee through both together. For
if only the red-making rays pafs through one Liquor,,
and only the blue-making through the other, no rays
can pafs through both. This Mr. Hook tried cafually
with Glafs-wedges filled with red and blue Liquors,
and was furprized at the unexpected event, the reafon
of it being then unknown ; which makes me truft the
more to his Experiment, though I have not tryed it
my felf. But he that would repeat it, muft take care
the Liquors be of very good and full Colours*

S a

Now

[; 4 °]

Now whilft Bodies become coloured by reflecting or
tranfmitting this or that fort of rays more copioufly than
the reft, it is to be conceived that they ftop and ftifle in
themfelves the rays which they do not refledtor tranfmit.
For if Gold be foliated and held between your Eye and
the Light, the Light looks blue, and therefore maffy Gold
lets into its Body the blue-making rays to be refledted
to and fro within it till they be ft opt and ftifled, whilft
it refledts the yellow-making outwards, and thereby
looks yellow. And much after the fame manner that
Leaf-gold is yellow by refledted, and blue by tranfmit-
ted Light, and maffy Gold is yellow in all politions of
the Eye; there are l'ome Liquors as the tindture of
Lignum JSTephritiam , and iome forts of Glai's which
tranfmit one fort of Light mod copioufly, and refiedt
another fort, and thereby look of feveral Colours, ac¬
cording to the polition of the Eye to the Light. But if
thele Liquors or Glaffes were lb thick and maffy that
no Light could get through them, I queftion not but
that they would like-all other opake Bodies appear of
one and the fame Colour in all politions of the Eye,
though this I cannot yet affirm by experience. For all
coloured Bodies, fo far as my Oblervation reaches, may
be feen through if made fufficiently thin, and therefore
are in fome meafure tranlparent, and differ only in de¬
grees of tranfparency from tinged tranfparent Liquors3
thefe Liquors, as well as thole Bodies, by a fumcient
thicknefs becoming opake. A tranlparent Body which
looks of any Colour by tranfmitted Light, may aifo
look of the lame Colour by refledted Light, the Light
of that Colour being refledted by the further furface of
the Body, orby the Air beyond it. And then the t e-
fledted Colour will be diminiihed, and perhaps ceafe, by

making

[ H 1 ]

making the Body very thick, and pitching it on the
back-fide to diminiih the reflexion of its further furface,
io that the Light reflected from the tinging particles
may predominate. In fuch cafes, the Colour of the re¬
flected Light will be apt to vary from that of the Light
tranfmitted. But whence it is that tinged Bodies and
Liquors reflect lbme fort of rays, and intromit or trank
mit other forts, lhall be laid in the next Book. In this
Propofition I content my felf to have put it paft difpute,
that Bodies have fuch Properties, and thence appear
coloured.

PROP. XL P ROB. VI.

% mixing coloured Lights to compound a Beam of Light
of the fame Colour and Nature sjuith a Beam of the Suns-
direS Lght , and therein to experience the truth of the
foregoing Brogofit ions.

Let AB Gabc reprefent a Prifm by which the Sun's Fig-
Light let into a dark Chamber through the Hole F, may .
be refracted towards the Lens M N, and paint upon it
at p, q, r, s and t, the ufual Colours violet, blue, green,
yellow and red,, and let the diverging rays by the re-
fradtion of this Lens converge again towards X, and
there,by the mixture of all thole their Colours,compound:
a white according to what was fliewn above. Then let
another Prifm DEG deg, parallel to the former, be
placed at X, to refraft that white Light upwards to¬
wards Y. Let the refracting Angles of the Prilins,
and their diftances from the Lens be equal, fo that the
rays which, converged from the Lens towards X, and
without refraction, would there have eroded and. diver¬
ged again, may by the refraction of the fecond Prifm be.

reduced

[ H 2 J

reduced into Parallelifm and diverge no more. For
then thofe rays will recompofe a Beam of white Light
X Y, If the refradling Angle of either Prifm be the
bigger, that Prifm mull be fo much the nearer to the
Lens. You will know when the Prifms and the Lens
are well fet together by obferving if the Beam of Light
X Y which comes out of the fecond Prifm be perfectly
white to the very edges of the Light, and at all diftan-
ces from the Prifm continue perfedlly and totally white
like a Beam of the Sun’s Light. For till this happens,
the politico of the Prifms and Lens to one another mull
be corredled, and then if by the help of a long Beam of
Wood, as is reprefented in the Figure, or by a Tube,
or fome other fuch inftrument made for that purpofe,
they be made fall in that fituation, you may try all the
fame Experiments in this compounded Beam of Light-
X Y, which in the foregoing Experiments have been
made in the Sun’s diredt Light. For this compounded
Beam of Light has the fame appearance, and is endowed
with all the fame Properties with a diredt Beam of the
Sun’s Light, fo far as my Obfervation reaches. And in
trying Experiments in this Beam you may by flopping
any of the Colours p, q, r, s and t, at the Lens, fee how
the Colours produced in the Experiments are no other
than thofe which the rays had at the Lens before they
entered the compolition ot this Beam : And by confe-
queued that they arife not from any new modifications
of the Light by refradtions and reflexions, but from the
various feparations and mixtures of the rays originally
endowed with their colour-making qualities.

So, for Alliance, having with a Lens \\ Inches broad,
and two Prifms on either Hand 6~ Feet diftant from the
Lens, made fuch a Beam of compounded Light: to

examin

C 1 43 ]

examin the reafon of the Colours made by Prifrns, I
refraded this compounded Beam of Light X Y with
another Prilfn HIK k h, and thereby eaft the ufual prift
matick Colours PQRST upon the Paper LV placed be¬
hind, And then by flopping any of the Colours p, q,
r, s, t, at the Lens, I found that the fame Colour would
vanifh at the Paper. So if the purple P was flopped at
the Lens, the purple P upon the Paper would vanifh,
and the reft of the Colours would remain unaltered,
unlefs perhaps the blue, fo far as fome purple latent in
it at the Lens might be feparated from it by the foL
lowing refradions. And fo by intercepting the green;
upon the Lens, the green R upon the Paper would va-
nilh, and fo of the reft ; which plainly fhews, that as
the white Beam of Light X Y was compounded of fe~
ve Lights varioufly coloured at the Lens, fo the Co¬
lours which afterwards emerge out of it by new refra-
dions are no other than thofe of which its whitenefs
was compounded. Therefradion of the Prifm HI K
kh generates the Colours PQRST upon the Paper,
not by changing the colorific qualities of the rays, but
by feparating the rays which had the very fame colorific
qualities before they entered the compofition of the re-
fraded Beam white of Light XY. For otherwife the rays
which were of one Colour at the Lens might be of ano¬
ther upon the Paper, contrary to what we find*

So again, to examin the reafon of the Colours of na^
tural Bodies, I placed fuch Bodies in the Beam of Light
XY, and found that they all appeared there of thofe
their own Colours which they have in Day-light, and
that thofe Colours depend upon the rays which had the:
lame Colours at the Lens before they, entred the compo¬
fition.

[ H4- ]

fition of that Beam. Thus, for inftance,Cinnaber illumi¬
nated by this Beam appears of the fame red Colour as in
Day-light • and if at the Lens you intercept the green-
making and blue-making rays, its rednefs will become
more full and lively : But if you there intercept the red-
making rays, it will not any longer appear red, but be¬
come yellow or green, or of ibme other Colour, accor¬
ding to the forts of rays which you do not intercept.
So Gold in this Light X Y appears of the fame yellow
Colour as in Day-light, but by intercepting at the Lens a
due quantity of the yellow-making rays it will appear
white like Silver (as I have tryed) which fhews that its
yellownefs arifes from the excels of the intercepted rays
tinging that whitenefs with their Colour when they are
let pals. So the infulion of Lignum JSfef hriticum (as I
have alfo tryed) when held in this Beam of Light X Y,
looks blue by the reflected part of the Light, and yellow
by the tranfmitted part of it, as when Yis viewed in Day¬
light, but if you intercept the blue at the Lens the infu¬
lion will lofe its reflected blue Colour, whilft its tranf-
rnitted red remains perfect and by the lofs of fome blue-
making rays wherewith it was allayed becomes morein-
tenfe and full. And, on the contrary, if the red and orange¬
making rays be intercepted at Lens, the infulion will
lofe its tranfmitted red, whilft its blue will remain and
become more full and perfect. Which fhews, that the in¬
fulion does not tinge the rays with blue and yellow, but
only tranfmit thole moft copiouily which were red-ma¬
king before, and reflects thole moft copiouily which were
blue-making before. And after the fame manner may the
realons of other Phenomena be examined, by trying
them in this artificial Beam of Light X Y.

THE

Book I. Part B. Plate 1

Book I. Part E. Plate I.

Book 1 , Part H Plate H.

-BookI. Part CL Plate E.

BookL Part K. Plate IE.

i m

cr'

T _

* 7 &7U

' ^

:: '

■

■ : .j

- f ' s [

■

Book I. Part R. Plate %

Book L Part E.J'kte b",

THE

SECOND BOOK

OPTICKS.

PART I.

Observations concerning the Reflexions , Refratlionsy and
Colours of thin tranjfarent Bodies .

I T has been obferved by others that tranfparent
Subftances, as Giafs, Water, Air, l$c. when made
very thin by being blown into Bubbles, or otherwife
formed into Plates, do exhibit various Colours accor¬
ding to their various thinnefs, although at a greater
thicknefs they appear very clear and colourlefs. In
the former Book I forbore to treat of thefe Colours,
becaufe they feemed of a more difficult coniideration,
and were not neceffary for eftabliffiing the Properties
of Light there difcourfed of. But becaufe they may
conduce to further difcoveries for completing the
Theory of Light, efpecially as to the conftitution of
the parts of natural Bodies, on which their Colours or
I ranlparency depend; I have here let down an ac¬
count of them. To render this Difcourfe fjiort and
diftindd, I have firft defcribed the principal of my

A a ' ' Obfer-

[ 2 3

Obfervations, and then coniidered and made ufe of
them. The Obfervations are thefe.

O B 3. I.

Compreiiing two Prifms hard together that their
Sides (which by chance were a very little convex)might
fomewhere touch one another: I found the place in
which they touched to become abfolutely tranfparent,
as if they had there been one continued piece of Glafs.
For when the Light fell fo obliquely on the Air, which
in other places was between them,as to be all reflected ;
it feemed in that place of contact to be wholly tranf-
mitted, infomuch that when looked upon, it appeared
like a black or dark Spot, by reafon that little or no
fenfible Light was reflected from thence, as from other
places; and when looked through it feemed (as it were)
a hole in that Air which was formed into a thin Plate,
by being compreffed between the Glades. And through
this hole Objects that were beyond might be feen di-
ftinCtly, which could not at all be feen through other
parts of the Glaffes where the Air was interjacent. Al¬
though the Glaffes were a little convex, yet this trans¬
parent Spot was of a conflderable breadth,which breadth
feemed principally to proceed from the yielding inwards
of the parts of the Glaffes, by reafon of their mutual
preffure. For by preffing them very hard together it
would become much broader than other wife.

OBS.

[ 9 ]

O B S. II.

When the Plate of Air, by turning the Prifms about
their common Axis, became fo little inclined to the in¬
cident Rays, that fome of them began to be tranfmit-
ted, there arofe in it many {lender Arcs of Colours
which at firft were (haped almoft like the Conchoid,
as you fee them delineated in the firft Figure, Arid Fig*
by continuing the motion of the Prifms, thefe Arcs in-
creafed and bended more and more about the faid trans¬
parent Spot, till they were completed into Circles or
Rings incompaffing it, and afterwards continually grew
more and more contracted.

Thefe Arcs at their firft appearance were of a violet f
and blue Colour, and between them were white Arcs
of Circles, which prefently by continuing the motion of
the Prifms became a little tinged in their inward Limbs
with red and yellow, and to their outward Limbs the \\
blue was adjacent. So that the order of thefe Colours
from the central dark Spot, was at that time white,
blue, violet ; black; red, orange, yellow, white, blue,
violet, teV. But the yellow and red were much fainter
than the blue and violet.

The motion of the Prifms about their Axis being con¬
tinued, thefe Colours contracted more and more,(brink-
ing towards the whitenefs on either fide of it, until they
totally vaniftied into it. And then the Circles in thole
parts appeared black and white, without any other Co¬
lours intermixed. But by further moving the Prifins
about, the Colours again emerged out of the whitenefs,
the violet and blue as its inward Limb, and at its out-

A -a a ward

ward Limb the red and yellow. So that now their order
from the central Spot was white, yellow, red; black ;
violet, blue, white, yellow, red, &c. contrary to what
it was before.

O B S. III.

When the Rings or fome parts of them appeared only
black and white, they were very diftinft and well de¬
fined, and the backnefs feemed as intenfe as that of
the central Spot. Alfo in the borders of the Rings,
where the Colours began to emerge out of the whiter
nefs, they were pretty diftind, which made them vi-
fible to a very great Multitude. I have fometimes
numbred above thirty Succeffions ( reckoning every
black and white Ring for one Succeffion) and feen
more of them, which by reafon of their final nefs I could
not number. But in other Petitions of the Prifms, at
which the Rings appeared of many Colours, I could not
diftinguifh above eight or nine of them, and the exte¬
rior of thofe were very eonfufed and dilute.

In thefe two Obfervations to fee the Rinps diftindh
and without any other Colour than black and white,!
found it neceffary to hold my Eye at a good diflance
from them. For by approaching nearer, although in the
fame inclination of my Eye to the plane of the Rings^
there emerged a blueifh Colour out of the white,
which by dilating it felf more and more into the black
rendred the Circles lefs diftindt, and left the white a
little tinged with red and yellow. I found alfo by
looking through a flit or oblong hole , which was
narrower than the Pupil of my Eye, and held dole to

it parallel to the Prifms, I could fee the Circles much
diftindter and vifible to a far greater number than
otherwife.

O B S. IV.

To obferve more nicely by the order of the Colours
which arofe out of the white Circles as the Rays be¬
came lefs and lefs inclined to the plate of Air; 1 took
two Object Glaffes, the one a Plano-convex for a four¬
teen-foot Telefcope, and the other a large double con¬
vex for one of about fifty-foot; and upon this,laying the
other with its its plane-fide downwards, I prefled them
flowly together,to make the Colours fuccefftvely emerge
in the middle of the Circles, and then flowly lifted
the upper Glafs from the lower to make them fuccef¬
fively vanifh again in the fame place. The Colour,,
which by prefling the Glaffes together emerged laft in
the middle of the other Colours, would upon its ftrft
appearance look like a Circle of a Colour almoft uni¬
form from the circumference to the center, and by
comprefling the Glaffes ftill more, grow continually
broader until a new Colour emerged in its center, and
thereby it became a Ring encompaffing that new Co¬
lour. And by comprefling the Glaffes ftill more, the
Diameter of this Ring would encreafe, arid the breadth
of its Orbit or Perimeter decreafe until another new*
Colour emerged in the center of the laft: And fo on
until a third, a fourth, a fifth, and other following
new Colours fucceffively emerged there, and became
Rings encompaffing the innernloft Colour]' the laft of.
which was the black Spot. And] on the contrary, by

lifting up the upper Glafs from the lower, the diameter
of the Rings would decreafe, and the breadth of their
Orbit encreafe, until their Colours reached fucceflively
to the center ; and then they being of a confiderable
breadth, I could more eafily difcern and diftinguifli
their Species than before. And by this means I ob-
ferved their Succeffion and Quantity to be as fol-
loweth.

Next, to the pellucid central Spot made by the con-
tafit of the Glaffes fucceeded blue, white, yellow, and
red, the blue was fo little in quantity that I could not
difcern it in the circles made by the Prifms, nor could
I well diftinguifli any violet in it, but the yellow and
red were pretty copious, and feemed about as much
in extent as the white, and four or five times more
than the blue. The next Circuit in order of Colours

immediately encompaffing thefie were violet, blue,
green, yellow, and red, and thefe were all of them co¬
pious and vivid, excepting the green, which was very
little in quantity, and feemed much more faint and
dilute than the other Colours. Of the other four, the
violet was the leaft in extent, and the blue lefs than
the yellow or red. The third Circuit or Order was
purple, blue, green, yellow, and red 5 in which the
purple feemed more reddifh than the violet in the
former Circuit, and the green was much more confpi-
cuous, being as brifque and copious as any of the other
Colours, except the yellow; but the red began to be
a little faded, inclining very much to purple. After
this fucceeded the fourth Circuit of green and red. The
green was very copious and lively, inclining on the one
fide to blue, and on the other fide to yellow. But in

this

Gy] ,

this fourth Circuit there was neither violet, blue, nor
yellow, and the red was very imperfect and dirty,
Alfo the fucceeding Colours became more and more im¬
perfect and dilute, till after three or four Revolutions
they ended in perfect whitenefs. Their Form, when the
Glaffes were molt compreffed fo as to make the black
Spot appear in the Center, is delineated in the Second
Figure; where «, r, n', e o, /a, i, h : /, w,«, o, p? 7, r: Fig.
j, t: V,x:y denote the Colours reck’ned in order from
the center, black, blue, white, yellow, red : violet,
blue, green, yellow, red : purple, blue, green, yellow,
red : green, red : greenilh blue, red : greenilh blue,
pale red : greenilh blue, reddilh white.

O B S. V.

To determine the interval of the Glaffes, or thick-
nefs of the interjacent Air, by which each Colour was
produced, I meafured the Diameters of the firft fix \\
Rings at the moil lucid part of their Orbits, and fqua-
ring them, I found their Squares to be in the Arith¬
metical Progreffion of the odd Numbers, 1.3.5.7.9.11,

And fince one of thefe Glaffes was Plain, and the other
Spherical, their Intervals at thofe Rings muft be in the
fame Progreffion. I meafured alfo the Diameters of
the dark or faint Rings between the more lucid Co¬
lours, and found their Squares to be in the Arithme¬
tical Progreffion of the even Numbers, 2.4. 6. 8.10. 1 a.

And it being very nice and difficult to take thefe mea-
fures exactly ; I repeated them at divers times at divers
parts of the Glaffes, that by their Agreement I might
be confirmed in them. And the fame Method I ufed in

deter-

[ 8 ]

determining fome others of the following Obfem^
tipns.

OBS, VI.

The Diameter of the fixth Ring at the mod lucid
part of its Orbit was ^ parts of an Inch, and the Dia¬
meter of the Sphere on which the double convex Ob-
jeCLGlafs was ground w r as about 102 Feet, and hence
I gathered the thicknefs of the Air or Aereal Interval
of the Glades at that Ring. But fome time alter, fuf-
pefting that in making this Obfervation 1 had not de¬
termined the Diameter of the Sphere with fuffident ac-
curatenefs, and being uncertain whether the Plano¬
convex Glafs was truly plain, and not fomething con¬
cave or convex on that fide which I accounted plain;
and whether I had not prefied the Glades together, as

I often did, to make them touch (for by preding Inch
Glades together their parts eafily yield inwards, and
the Rings thereby become fendbly broader than they
would be, did the Glades keep their Figures) I re¬
peated the Experiment , and found the Diameter of
the fixth lucid Ring about parts of an Inch. I re¬
peated the Experiment alio with fuch an ObjeCt-Glais
of another Telefcope as I had at hand. This was a double
convex ground on both fides to one and the lame
Sphere, and its Focus was diftant from it 8^ Inches.
And thence, if the Sines of incidence and refraction of
the bright yellow Light be affumed in proportion as

II to i y, the Diameter of the Sphere to which the
Glafs was figured will by computation be found 18 2 In¬
ches. This Glafs I laid upon a fiat one, fo that the

1 black

black Spot appeared in the middle of the Rings of Colours
without any other preffure than that of the weight of
the Glafs. And now meafuring the Diameter of the
fifth dark Circle as accurately as 1 could, I found it the
fifth part of an Inch precifely. This meafure was taken
with the points of a pair of Compaffes on the upper fui>
face on the upper Glafs, and my Eye was about eight
or nine Inches diftance from the Glafs, almoft perpen¬
dicularly over it, and the Glafs w T as \ of an Inch thick,
and thence it is eafy to colled! that the true Diameter
of the Ring between the Glaffes was greater than its
meafured Diameter above the Glaffes in the proportion
of 80 to 79 or thereabouts, and by confequence equal
to ~ parts of an Inch, and its true Semi-diameter equal
to ~ parts. Now as the Diameter of the Sphere (18a In¬
ches) is to the Semi-diameter of this fifth dark Ring
( — parts of an Inch) io is this Semi-diameter to the 0
thicknefs of the Air at this fifth dark Ring; which fe
therefore or mm P arts an Inch, and the fifth
part thereof; viz. the ^ 7 th part of an Inch, is the
thicknefs of the Air at the firft of thefe dark Rings;

The fame Experiment { repeated with another dou¬
ble convex ObjedTgiafs ground on both fides to one and
the fame Sphere. Its Focus was diftant from it 168-
Inches, and therefore the Diameter of that Sphere was
184 Inches. This Glafs being laid upon the fame
plain Glafs, the Diameter of the fifth of the dark
Rings, when the black Spot in their center appeared
plainly without preffing the Glaffes, was by the mea-
fure of the Compafies upon the upper Glais im parts
of an Inch, and by confequence between the Glaffes it
was|~, For the upper Glafs was § of an Inch thick,

B b and

and my Eye was diftant from it 8 Inches. And a third
proportional to half this from the Diameter of the
Sphere is ^ parts of an Inch. This is therefore the
thicknefs of the Air at this Ring, and a fifth part there¬
of, viz. the g^-th part of an Inch is the thicknefs there¬
of at the firft of the Rums as above.

o ,

I tryed the fame thing by laying thefe Objeft-Glaffes
upon Hat pieces of a broken Looking-glafs, and found
the fame meafures of the Rings : Which makes me
rely upon them till they can be determined more ac¬
curately by Glades ground to larger Spheres, though
in fuch Glafies greater care muft be taken of a true

nil.

Thefe Dimen fior is were ta

n when my Eye was

placed almoft perpendicularly over the Glades, being
about an Inch, or an Inch and a quarter, diftant from
° the incident rays, and eight Inches diftant from the
Glafs; fo that the rays were inclined to the Glafs in an
Angle of about \ degrees. Whence by the following
Observation you will underftand, that had the rays
been perpendicular to the Glades, the thicknefs of the
Air at thefe Rings would have been lefs in the propor¬
tion of the Radius to the lecant of 4 degrees, that is of
loooo. Let the thicknedes found be therefore dimi-
nilhed in this proportion, and they will become and
or ( to ufe the neareft round number) the 8 ~th
part of an Inch. This is the thicknefs of the Air at the
darkeft part of the firft dark Ring made by perpendi¬
cular rays, and half this thicknefs multiplied by the
progredion,i,3,5,7,,9,1 i,l 5>c. gives the thicknedes of the
Air at the mod: luminous parts of all the brighteft

Rings, vk. ^3

i7C)OO05- IJtiQCQ) IJijCCO? *70000?.

i 9 V\ their arithmetical

means.

[II]

means — 0 c. being its thiekneffes at the

darkeft parts of all the dark ones.

obs, vir.

The Rings were leaft when my Eye was placed per¬
pendicularly over the Glaffes in the Axis of the Rings :
And when I viewed them obliquely they became big¬
ger, continually fwelling as I removed my Eye further
trom the Axis. And partly by meafuring the Diameter
ol the fame Circle at feveral obliquities of my Eye,
partly by other means, as alio by making ule of the
two Frilms for very great obliquities. I found its Dia¬
meter, and confequently the thicknefs of the Air at its
perimeter in all thole obliquities to be very nearly in the
proportions expreffed in this Table.

Angle

cidence

Air,

of In-
on the

Angle of Re¬
fraction into
the Air .

Diameter of
the Ring,

Thicknefs of
the Air .

deg.

min.

f. 1 >

*°iS

Ic i

I0 i .

3°

IO ;

2 4

3°

4°

I l|

151 I

2 9

5 °

5 8

6 5

*5?

l 9

16 |

J 94

2 7

22f

5 ^

2 9

84 ,-; 1

OO j

I 2 2l

[ 12 ]

In th&tfco firftColumns are e&preffed the obliquities
of the incident and emergent rays to the plate of the
Air, that is, their angles of incidence and refraction. In
the third Column the Diameter of any coloured Ring
at thole obliquities is expreffed in parts, of which ten
conftitute that Diameter when the rays are perpendicu¬
lar. And in the fourth Column the thicknefs of the Air
at the circumference of that Ring is expreffed in parts
of which alio ten conftitute that thicknefs when the rays
ate perpendicular..

And from thefe meafures I feem to gather this Rule :
That the thicknefs of the Air is proportional to the fe-
cantwof an angle, whole Sine is a certain mean proper-
tionai between the Sines of incidence and refraCtionr
And that mean proportional, lb far as by thefe meafures
I can determine it, is the firft of an hundred and fix
arithmetical mean proportionals between thofe Sines
counted from the Sine of refraction when the refra¬
ction is made out of the Glafs into the plate of Air, or
from the Sine of incidence when the refraCtion is
made out of the plate of Air into the Glafs.

| • J I j - l | . O O C | (*-j- - i f

-• 1 ‘ O B S. ¥ 11 L V ; - -

\ 1 ! ^ i 1 O ay : ; *

The dark Spot in the middle of the Rings increafed
alfo by the obliquation of the Eye, although almoft in-
fenlibly. But if inftead of the Object-Glades the Prifms
were madeufeof, itsincreafe was more manifeft when
¥iewed fo obliquely that no Colours appeared about it.
It was leaft when the rays were incident moft obliquely
on the interjacent Air, and as the obliquity decreafed
it increafed more and more until the coloured Rings ap-
hi peared.

peared, and then decreafed again, 1 but not fo much as
it increafed before. And hence it is evident, that the
tranfparency was not only at the abfolute contact of the
Glaffes, but alfo where they had feme little interval.
I have fometimes obferved the Diameter of that Spot to
be between half and two fifth parts of the Diameter of
the exterior circumference of the red in the firft cir¬
cuit or revolution of Colours when viewed almoft per¬
pendicularly ; whereas when viewed obliquely it hath
wholly vanifhed and become opake and white like the
other parts of theGlafs; whence it may be collected
that the Glaffes did then fcarcely, or not at all, touch
one another, and that their interval at the perimeter
of that Spot when viewed perpendicularly was about a
fifth or fixth part of their interval at the circumference
of the faid red.

O B S. IX

. '' f . { i , V f u - ’ ' . t

• - - 'jj'..! : ■' „ ; ' J • f\ : * : . i :■ - " •; V; ; *; , ; -■

By ■ looking through the two contiguous Obje£L
Glaffes, I found that the interjacent Air exhibited Rings
of Colours, as well by tranfmitting Light as by reflect¬
ing it. The central Spot was now white, and from it
the order of the Colours were yellowife red; black ;
violet, blue, white, yellow, red ; violet, blue, green,
yellow, red, bv. But thefe Colours were very faint
and dilute unlefs when the Light was trajefted very
obliquely through the Glaffes : For by that means they
became pretty vivid. Only the fir ft yellowife red, like
the blue in tire fourth Obfervation, was fo little and
faint as fcarcely to be difcerned. Comparing the co¬
loured Rings made by reflexion , with thefe made by

tranff

[ 1 +]

tranfmiffion of the Light ; I found that white was op-
polite to black, red to blue, yellow to violet, and green
to a compound of red and violet. That is, thofe parts
of the Glals were black when looked through, which
when looked upon appeared white, and on the con¬
trary. And fo thofe which in one cafe exhibited blue,
did in the other cafe exhibit red. And the like of the
Fig. 5. other Colours. The manner you have reprefen ted in
the third Figure, where AB, CD, are the furfaces of
the daffes contiguous at E, and the black lines be¬
tween them are their diffances in arithmetical progrel-
lion, and the Colours written above are feen by re¬
flected Light, and thofe below by Light tranfmitted.

O B S. X.

Wetting the ObjeCt-Glaffes a little at their edges,
the water crept in (lowly between them, and the Cir¬
cles thereby became lels and the Colours more ^ faint :
Infomuch that as the water crept along one half of
them at which it firft arrived would appear broken off
from the other half, and contracted into a lets room.
By meafuring them I found the proportions of their
Diameters to the Diameters of the like Circles made by
Air to be about leven to eight, and conlequently the in¬
tervals of the Glaffes at like Circles, cauled by thofe
two mediums Water and Airfare as about three to four.
Perhaps it may be a general Rule, That if any other
medium more or lefs denfe than water be compreffed
between the Glaffes, their intervals at the Rings caufed
thereby will be to their intervals caufed by interjacent

[!$]

Air, as the Sines are which meafure the refraction made
out of that medium into Air.

OBS. XI.

When the water was between the Glalfes, if I pref-
fed the upper Glafs varioufly at its edges to make the*
Rings move nimbly from one place to another, a little
white Spot would immediately follow the center of
them, which upon creeping in of the ambient water
into that place would prefently vanifh. Its appearance
was fuch as interjacent Air would have caufed, and it
exhibited the fame Colours. But it was not Air, for
where any bubbles of Air were in the water they would
not vanifh. The reflexion mult have rather been caufed
by a fubtiler medium,, which could recede through the
Glalfes at the creeping in of the water.

O B S. XIL

Thefe Obfervations were made in the open Air. But-
further to examin the effects of coloured Light falling
on the Glalfes, I darkened the Room, and viewed them
by reflexion of the Colours of a Prifm call on a Sheet:
of white Paper,. my .Eye being fo placed that I cou ld
fee the coloured Paper by reflexion in the Glalfes, as
in a Looking-glafs. And by this means the Rings be¬
came diftinfter and vilible to a far greater number than
in the open Air. I have fometimes feen more than
twenty of them, whereas in the open Air I could not
difcern above eight or nine.

O B S. XIII.

Appointing an afliftant to move the Prifm to and
fro about its Axis, that all the Colours might fuccef-
fively fall on that part of the Paper which I faw by
reflexion from that part of the Glades, where the Cir¬
cles appeared, fo that all the Colours might be fuccef-
fively reflected front the Circles to my Eye whilft I held
it immovable, I found the Circles which the red Light
made to be manifeftly bigger than thofe which were
made by the blue and violet. And it was very plea-
fant to fee them gradually fwell or contradt according
as the Colour of the Light was changed. The inter¬
val of the Glades at any of the Rings when they were
made by the utmoft red Light, was to their interval at
the fame Ring when made by the utmoft violet, greater
than as 3 to a, and lefs than as 13 to 8,by the mod of my
Obfervations it was as 14 to 9. And this proportion
feemed very nearly the lame in all obliquities of my
Eye 3 unlefs when two Prifms were made ufe of inftead
of the Obje6f-Glades. : For then at a' certaimgreat
obliquity of my Eye, the Rings made by the feveral
Colours feemed equal, and at a greater obliquity thofe
made by the violet would be greater than the fame
Rings made by the red. The refraction of the Prifm
in this cafe caufing the mod: refrangible rays to fall
more obliquely on that plate of the Air than the lead:
refrangible ones. Thus the Experiment fucceeded in
the coloured Light, which was fufficiently ftrong and
copious to make the Rings fenflble. And thence it
maybe gathered, that if the moft refrangible and leaft

refran*

[ J 7 ]

refrangible rays had been copious enough to make the
Rings fenfible without the mixture of other rays, the
.proportion which here was 14 to 9 would have been a
little greater, fuppofe 14 ~ or 14 l to 9.

O B S. XIV.

Whilft the Prifm was turn'd about its Axis with an
uniform motion, to make all the feveral Colours fall
fucceffively upon the ObjeCt rGlaffes, and thereby to
make the Rings contract: and dilate : The contraction
or dilation of each Ring thus made by the variation of
its Colour was fwifteft in the red, and iloweft in the
violet, and in the intermediate Colours it had inter¬
mediate degrees of celerity. Comparing the quantity
of contraction and dilation made by all the degrees of
each Colour, I found that it was greateft in the red ;
lefs in the yellow, (till lefs in the blue, and leaft in the
violet. And to make as juft an eftimation as 1 could of the
proportions of their contractions or dilations, I obferved
that the whole contraction or dilation of the Diameter
of any Ring made by all the degrees of red, was to that
of the Diameter of the fame Ring made by all the de¬
grees of violet, as about four to three, or five to four, and
that when the Light was of the middle Colour between
yellow and green, the Diameter of the Ring was very
nearly an arithmetical mean between the greateft Dia¬
meter of the fame Ring made by the outmoft red, and
the leaft Diameter thereof made by the outmoft violet:
Contrary to what happens in the Colours of the oblong
SpeCtrum made by the refraCtion of a Prifm, where the
red is moft contracted, the violet moft expanded, and

D d m

[i8]

in the midft of all the Colours is the confine of green
and blue. And hence I feem to colled that the thick-
neffes of the Air between the Glaffes there., where the
Ring is lucceffively made by the limits of the five prin¬
cipal Colours (red, yellow, green, blue, violet) in order
(that is, by the extreme red, by the limit of red and
yellow in the middle of the orange, by the limit of
yellow and green, by the limit of green and blue, by
the limit of blue and violet in the middle of the in-
digo, and by the extreme violet) are to one another
very nearly as the fix lengths of a Chord which found
the notes in a fixth Major, fol, la , mi, fa, fol , la. But
it agrees fomething better with the Obiervation to fay,
that the thicknefies of the Air between the Glaffes there,
where the Rings are fucceffively made by the limits of
the feven Colours, red, orange, yellow, green, blue, in¬
digo, violet in order, are to one another as the Cube-
roots of the Squares of the eight lengths of a Chord,
which found the notes in an eighth, fol, la, fa, fol, la,
mi, fa, fol ; that is, as the Cube-roots of the Squares
of the Numbers, i, f, % \ \ \ ^ \.

O B S. XV.

Thefe Rings were not of various Colours like thole
made in the open Air, but appeared all over of that
prifmatique Colour only with which they were illu¬
minated. And by projecting the prifmatique Colours
immediately upon the Glaffes, I found that the Light
which fell on the dark Spaces which were between
the coloured Rings , was tranfmitted through the
Glaffes without any variation of Colour. For on a

white

[I 9 1

white Paper placed behind, it would paint Rings of

the fame Colour with thofe which were reflected, and

of the bignefs of their immediate Spaces. And from
thence the origin of thefe Rings is manifeft; namely,

That the Air between the Glaffes, according to its va¬
rious thicknefs, is difpofed in fome places to reflect,
and in others to tranimit the Light of any one Co¬
lour (as you may fee reprefented in the fourth Figure) fig, ^
and in the fame place to reflect that of one Colour
where it tranfmits that of another.

O B S. XVI. 1

The Squares of the Diameters of thefe Rings made
by any prilmatique Colour were in arithmetical pro-
greflion as in the fifth Obfervation. And the Diameter
of the fixth Circle, when made by the citrine yellow,
and viewed almoft perpendicularly, was about s ~ parts
of an Inch, or a little lefs, agreeable to the fixth Ob¬
fervation. ■

The precedent Obfervations were made with a rarer
thin medium, terminated by a denfer, fuch as was Air
or Water compreffed between two Glaffes. In thofe
that roliow are fet down the appearances of a denfer
medium thin’d within a rarer, fuch as are plates of
Muicovy-glafs, Bubbles of Water, and fome other thin
fubftances terminated on all fides with Air. -

0 B 3.

, Dd a

O B S. XVII.

If a Bubble be blown with Water firfl: made tenacious
by diffolving a little Soap in it, ? tis a common Obfer¬
vation, that after a while it will appear tinged with a
great variety of Colours, To defend thefe Bubbles
from being agitated by the external Air (whereby their
Colours are irregularly moved one among another, fe
that no accurate Obfervation can be made of them,) as
foon as I had blown any of them I covered it with a
clear Glafs, and by that means its Colours emerged in
a very regular order, like fo many concentrick Rings
incompaffing the top of the Bubble. And as the
Bubble grew thinner by the continual fubfiding of the
Water, thefe Rings dilated llowly and over-fpread the
whole Bubble, defending in order to the bottom of it,
where they vanifhed fucceffively. In the mean while,
after all the Colours were emerged at the top, there
grew in the Center of the Rings a fmall round black
Spot, like that in the firfl: Obfervation, which conti¬
nually dilated it felf till it became Fometimes more than
- or f of an Inch in breadth before the Bubble broke.
At firfl: I thought there had been no Light refledted from
the Water in that place, but obferving it more cu-
rioufly, I faw within it levend fmaller round Spots,
which appeared much blacker and darker than the reft,
whereby I knew that there was fome reflexion at the
other places which were not fo dark as thofe Spots.
And by further tryal I fouud that I could fee the Images
©f fome things (as of a Candle or the Sun ) very faint¬
ly reflected, not only from the great black Spot, but

alfo

[21 ]

aifo from the little darker Spots which were with¬
in it.

Befides the aforefaid coloured Rings there would
often appear fmall Spots of Colours, afcending and de¬
fending up and down the fides of the Bubble, by reafon
of feme inequalities in the fubfiding of the Water.
And fometimes fmall black Spots generated at the fides
would afcend up to the larger black Spot at the top of
the Bubble, and unite with it.

OBS. XVIIL

Becaufe the Colours of thefe Bubbles were more ex¬
tended and lively than thofe of the Air thin’d between
two Glafles, and fo more eafy to to diftinguiftied, I
fhall here give you a further defcription of their order,,
as they were obferved in viewing them by reflexion of
the Skies when of a white Colour, whilft a black Sub-
ftance was placed behind the Bubble. And they were
thefe, red, blue; red, blue ; red, blue • red, green ;
red, yellow, green, blue, purple; red, yellow,, green,
blue, violet; red, yellow, white, blue, black.

The three firft Succeffions of red and blue were very
dilute and dirty, efpecialiy the firft, where the red
feemed in a manner to be white^ Among thefe there
was fcarce any other Colour fenfible befides red and
blue, only the blues ( and principally the fecond blue
inclined a little to green.

The fourth red was alfo dilute and dirty, but not
fo much as the former three; after that fucceeded little,
or no yellow, but a copious green, which at firft incli¬
ned a little to yellow, and then became a pretty brifque

and

. [ M ]

and good willow green, and afterwards changed to a
bluifh Colour; but there fucceeded neither blue nor
violet.

The fifth red at firft inclined very much to purple,
and afterwards became more bright and brifque, but
yet not very pure. This was fucceeded with a very
bright and intenfe yellow, which w r as but little in
quantity, and foon changed to green : But that green
was copious and fomething more pure, deep and lively,
than the former green. After that followed an excel¬
lent blue of a bright sky-colour, and then a purple,
which was lefs in quantity than the blue, and much
inclined to red. /

The fixth Red was at firft of a very fair and lively
Scarlet, and foon after of a brighter Colour, being
very pure and brifque, and the beft of all the
reds. Then after a lively orange followed an intenfe
bright and copious yellow, which was alfo the beft
of all the yellows, and this changed firft to a greenifti
yellow, and then to a greenifti blue ; but the green
between the yellow and the blue, was very little and
dilute, fceming rather a greenifti white than a green.
The blue which fucceeded became very good, and of a
very fair bright sky-colour, but yet fomething inferior
to the former blue; and the violet was intenfe and
deep with little or no rednefs in it. And lefs in quan¬
tity than the blue.

In the laft red appeared a tinfture of fcarlet next
to violet, which foon changed to a brighter Colour,
inclining to an orange; and the yellow which followed
was at firft pretty good and lively, but afterwards it
grew more dilute, until by degrees it ended in perfedt

white-

(

BooTc,n.

Plate, H.

03 ] ,

whitenefs. And this whitenefs, if the Water was very
tenacious and well-tempered, would ftowly fpread and
dilate it felf over the greater part of the Bubble; con¬
tinually growing paler at the top, where at length it
would crack in many places, and thofe cracks, as they
dilated, would appear of a pretty good, but yet obfcure
and dark sky-colour j the white between the blue Spots
diminiftiirig, until it refembled the threds of an irre¬
gular Net-work, and foon after vanifhed and left all
the upper part of the Bubble of the laid dark blue
Colour. And this Colour, after the aforefaid manner,
dilated it felf downwards , until fometimes it hath
overfpread the whole Bubble. In the mean while at
the top, which was of a darker blue than the bottom,
and appeared alfo full of many round blue Spots, feme-
thing darker than the reft, there would emerge one
or more very black Spots, and within thofe other Spots
of an interifer blacknefs, which I mentioned in the
former Obfervation j and thefe continually dilated
themfelves until the Bubble broke.

If the Water was not very tenacious the black Spots
would break forth in the white, without any fenlible
intervention of the blue. And fometimes they would
break forth within the precedent yellow, or red, or
perhaps within the blue of the fecond order, before
the intermediate Colours had time to difplay them¬
felves.

By this defcription you may perceive how great an
affinity thefe Colours have with thofe of Air defcri-
bed in the fourth Obfervation, although fet down in
a contrary order, by reafon that they begin to appear
when the Bubble is thickeft , and are molt conve*

niently

niently reckoned from the loweft and thickeft part of
the Bubble upwards.

OBS, XIX.

Viewing in feveral oblique pofitions of my Eye
the Rings of Colours emerging on the top of the Bubble,
I found that they were fenfibly dilated by increafing
the obliquity, but yet not fo much by far as thofe
made by thin’d Air in the feventh Obfervation. For
there they were dilated fo much as, when viewed
moft obliquely, to arrive at a part of the plate more
than twelve times thicker than that where they ap¬
peared when viewed perpendicularly; whereas in this
cafe the thicknefs of the Water, at which they arrived
when viewed moft obliquely, was to that thicknefs
which exhibited them by perpendicular rays, fome-
thing lefs than as 8 to 5. By the beft of my Observations
it was between 15 and 15; to 10, an increafe about
ap times lefs than in the other cafe.

Sometimes the Bubble would become of an uniform
thicknefs all over, except at the top of it near the black
Spot, as I knew, becaufe it would exhibit the fame
appearance of Colours in all pofitions of the Eye. And
then the Colours which were feen at its apparent cir¬
cumference by the obliqueft rays, would be different
from thofe that were feen in other places, by rays lefs
oblique to it. And divers Spectators might fee the
fame part of it of differing Colours, by viewing it at
very differing obliquities. Now obferving how much
the Colours at the fame places of the Bubble, or at di¬
vers places of equal thicknefs, were varied by the

feveral

[* 5]

feveral obliquities of the rays; by the affiftance of the
4th, 14th, 16 th and 18 th Obfervations, as they are
hereafter explained, I colled the thicknefs of the Water
requifite to exhibit any one and the fame Colour, at fe¬
veral obliquities, to be very nearly in the proportion
expreffed in this Table.

Incidence on
the Water .

Refraction in -
to the Water,

Thicknefs of
the Water .

deg.

min.

deg.

min.

I I

l°5

3°

10 -;

3 2

4 °

3°

J 3

• Hi

2 5

_ In the two firft Columns are expreffed the obliqui¬
ties of the rays to the fuperficies of the Water, that
is, their Angles of incidence and refradion. Where
I fuppofe that the Sines which meafure them are in
round numbers as 3 to 4, though probably the diffo-
lution of Soap in the Water, may a little alter its
reiradive Vertue. In the third Column the thicknefs
of the Bubble, at which any one Colour is exhibited
in thofe feveral obliquities, is expreft in parts,of which
ten conftitute that thicknefs when the rays are perpen¬
dicular.

I have fometimes obferved, that the Colours which
arife on polilhed Steel by heating it, or on Bell-metal,
and fome other metalline fubftances, when melted and

E e poured

[2d]

poured on the ground , where they may cool in the
open Air, have, like the Colours of Water-bubbles,
been a little changed by viewing them: at divers ob¬
liquities, and particularly that a deep Hue, or violet,
when viewed very obliquely, hath been changed to a
deep red. But the changes of thefe Colours are not fo
great and fenfible as of thofe made by Water. For the
Scoria or vitrified part of the Metal, which rnoft Me¬
tals when heated or melted do continually protrude,
and fend out to their furface, and which by covering
the Metals in form of a thin glaffy skin, caules thefe
Colours, is much denfer than Water ; and I find that
the change made by the obliquation of the Eye is lead
in Colours of the aenfeft thin fubftances.

O B 3 . XX.

As in the ninth Obfervation, fo here, the Bubble, by
tranfmitted Light, appeared of a contrary Colour to
that which it exhibited by reflexion. Thus when the
Bubble being looked on by the Light of the Clouds re-
fltefited from it, feemed red at its apparent circumfe¬
rence, if the Clouds at the fame time, or immediately
after, were viewed! through it, the Colour at its cir¬
cumference would be blue; And, on the contrary,,
wftetr by reflected Light it appeared blue, it would ap¬
pear red by tranfmitted* Light.

O B S. XXL

By wetting vfery thin plates of Mufcovy-glafs, whofe
thinnefs made the like Colours appear,, the Colours

became

c27 ]

became mote faint and languid; especially by wetting
the plates on that fide oppofite to the Eye: But I could
not perceive any variation of their fpecies. So then
the thieknefs of a plate requifite to produce any Co¬
lour, depends only on the denfity of the plate, and
not on that of the ambient medium: And hence, by the
10th and 16th Oblervations, may be known the thick-
nefs which Bubbles of Water, or Plates of Mufcovy-
glafs, or other fubftances, have at any Colour pro¬
duced by them.

O B S. X X I I.

A thin tranfparent Body, which is denfer than Its
ambient medium, exhibits more brifque and vivid Co¬
lours than that which is fo much rarer * as I have
particularly obferved in the Air and Glafs. For blow¬
ing Glafs very thin at a Lamp-furnace, thofe plates
incompafled with Air did exhibit Colours much
more vivid than thofe of Air made thin between two
Glaffes.

O B S. XXIII.

Comparing the quantity of Light reflected from the
feveral Rings, I found that it was moft copious from
the firft or inmoft, and in the exterior Rings be¬
came gradually lefs and left. Alfo the whitenefs of
the firft Ring was ftronger than that reflected from
thofe parts of the thinner medium which were with¬
out the Rings; as I could manifeftly perceive by view¬
ing at a dirtance the Rings made by the two Object-

E e 3 Glafles ?

[28]

Glaffes | or by comparing two Bubbles of Water blown
at diftant times, in the fir ft of which the whitenefs
appeared, which fucceeded all the Colours, and in
the other, the whitenefs which preceded them all

OBS. XXIV.

When the two ObjeCt-Glaffes w r ere lay’d upon one
another, fo as to make the Rings of the Colours ap¬
pear, though with my naked Eye I could not difcern
above 8 or 9 of thofe Rings, yet by viewing them
through a Prifm I have feen a far greater multitude,
infomuch that I could number more than forty, befides
many others, that were fo very final! and dole toge¬
ther, that I could not keep my Eye fteddy on them
feverally fo as to number them, but by their extent I have
fometimes eftimated them to be more than a hundred.
And I believe the Experiment may be improved to the
difcovery of far greater numbers. For they feem to
be really unlimited, though vifible only fo far as they
can be feparated by the refraction, as I (hall hereafter
explain.

But it was but one fide of thefe Rings, namely, that
towards which the refraction was made, which by that
refraCtion was rendered diftinCt, and the other fide be¬
came more confufed than when viewed by the naked
Eye, infomuch that there I could not difcern above
one or two, and fometimes none of thofe Rings, of
which I could difcern eight or nine with my naked
Eye, And their Segments or Arcs, which on the
other fide appeared fo numerous, for the moll part

[ 2 9 ]

exceeded not the third part of a Circle. If the Re¬
fraction was very great, or the Prifm very diftant from
the ObjeCt-Glaffes, the middle part of thofe Arcs be¬
came alfo confufed, fo as to difappear and conftitute an
even whitenefs, whilft on either fide their ends, as alfo
the whole Arcs furtheft from the center, became di-
ftindter than before, appearing in the form as you fee
them defigned in the fifth Figure. Fig

The Arcs, where they feemed diftinCteft, were only
white and black fucceffively, without any other Co¬
lours intermixed. But in other places there appeared
Colours, whofe order was inverted by the refraCtion
in fuch manner, that if I firft held the Prifm very near
the Objedt-Glafles, and then gradually removed it
further off towards my Eye, the Colours of the ad,

3d, 4th, and following Rings ihrunk towards the white
that emerged between them, until they wholly va-
nifhed into it at the middle of the Arcs, and after¬
wards emerged again in a contrary order. But at
the ends of the Arcs they retained their order un¬
changed.

I have fometimes fo lay’d one Objedt-Glafs upon
the other, that to the naked Eye they have all over
feemed uniformly white, without the lead appearance-
of any of the coloured Rings - } and yet by viewing
them through a Prifm, great multitudes of thofe Rings
have difcovered themfelves. And in like manner plates,
of Mufcovy-glafs, and Bubbles of Glafs blown at a
Lamp-furnace, which were not fo thin as to exhibit
any Colours to the naked Eye, have through the Prifm
exhibited a great variety of them ranged irregu-
larly up and, down in the form of waves. And fo

Bubbles.

Babbles of Water, before they began to exhibit their
Colours to the naked Eye of a By-ftander, hare ap¬
peared through a Prifm, girded about with many pa¬
rallel and horizontal Rings; to produce which effe®,
it was neceflary to hold the Prifm parallel, or very
nearly parallel to the Horizon, and to difpofe it fo
that the rays might be refracted upwards,

-. ft

THE

C P ]

THE

SECOND BOOK

PART II.

Remarks ufon the foregoing Oifervations..

Aving given my Obfervations of thefe Colours,

_ before I make ufe of them to unfold the Caufes

of the Colours of natural Bodies, it is convenient that
by the fimpleft of them, fuch as are the ad, 3d, 4th,

9th, rath, 18th, 20th, and a4th, I firft explain the
more expounded. And firft to fhew how the Colours
in the fourth and eighteenth Obfervations are produ-
eed, let there be taken in any right line from the point
% the lengths ¥A, YB, Y C, YD, YE, YF, Y G y Fig.$
YH, in proportion to one another, as the Cube-roots
of the Squares of the numbers, ^ 3 , f, ? 1, where¬

by the lengths of a mufical Chord to found all the Notes 1
ins an Eighth are reprefented; that is, in the propor- >
tion of the numbers 630a, 6814, 7114, 7631, 8255,,

8855, 2243,: ioooo.. And at the points A, B, C, D

E,F.

CS 2 !

E, F, G, H, let perpendiculars A B bv. be ere&ed,
by whole intervals the extent of the feveral Colours
fet underneath againft them, is to be reprefented. Then
divide the line A « in fuch proportion as the numbers
i, 2, 3, 5, 6, 7, 9, io, 11 , 1 9V. fet at the points of divi¬
sion denote. And through thofe divilions from Y
draw lines i I, 2 K, 5 L, 5 M, 6 N, 7 O, be.

Now if A 2 be iuppofed to reprefent the thicknefs
of any thin transparent Body, at which the outmoft
violet is moft copioufly refie&ed in the firft Ring, or
Series of Colours, then by the 1 3th Obfervation H K,
will reprefent its thicknefs, at which the utmoft red
is moft copioufly refle&ed in the fame Series. Alfo
by the 5 th and 16th Obfervations, A 6 and HN will
denote the thickneffes at which thofe extreme Colours
are moft copioufly reflected in the fecond Series, and
A 1 o and H Q the thickneffes, at which they are
moft copioufly reflected in the third Series, and fo on.
And the thicknefs at which any of the intermediate
Colours are reflefted moft copioufly, will, according to
the 14th Obfervation, be defined by the diftance of the
line A H from the intermediate parts of the lines 2 K,
6N, 10Q, be. againft which the names of thofe Co¬
lours are written below. ; •

But further, to define the latitude of thefe Colours in
each Ring or Series, let A 1 defign the leaft thicknefs,
and A 3 the greateft thicknefs, at which the extreme
violet in the firft Series is reflected, and let H and
H L, defign the like limits for the extreme red, and
let the intermediate Colours be limited by the inter¬
mediate parts of the lines 1 1 , and 3 L, againft which
the names of thofe Colours are written, and fo on: But

yet

[ 33 ]

yet with this caution, that the reflections be fuppofed
ftrongeft at the intermediate Spaces, a K, 6 N, 10Q fee.
and from thence to decreafe gradually towards thefe li¬
mits, 11, 3 L, 5 M, 7O, he. on either fide; where
you mull not conceive them to be precifely limited,
but to decay indefinitely. And whereas I have afiigned
the fame latitude to every Series, I did it, becaufe al¬
though the Colours in the firft Series feem to be a little
broader than the reft, by reafon of a ftronger reflexion
there, yet that inequality is fo infenfible as fcarcely to
be determined by Oblervation.

Now according to this defeription, conceiving that
the rays originally of feveral Colours are by turns re¬
flected at the Spaces 11 L 3, 5 M O 7, 9 P R 11, he.
and tranfmitted at the Spaces AHIi,3LM5,70P9,
he. it is eafy to know what Colour muft in the open Air
be exhibited at any thicknefs of a tranfparent thin body.
For if a Ruler be applied parallel to A H, at that di-
ftance from it by which the thicknefs of the body is
reprefented, the alternate Spaces 11L 3, 5 M O 7,35V.
w r hich it croffeth will denote the reflected original Co¬
lours, of which the Colour exhibited in the open Ay:
is compounded. Thus if the conftitution of the green
in the third Series of Colours be defired, apply the
Ruler as you fee at * e * and by its palling through
fome of the blue at * and yellow at 0 ', as well as through
the green at ? , you may conclude that the green exhi¬
bited at that thicknefs of the body is principally con-
ftituted of original green, but not without a mixture
of fome blue and yellow',

Ff. By

C34]

$y this means you may know how the Colours front
the center of the Rings outward ought to fucceed in
order as they were defcribed in the 4th and 18th Ob-
fervations. For if you move the Ruler gradually from
AH through all diftances, having paft over the firft
fpa.ee which denotes little or no reflexion to be made
by thinneft fubftances, it will firft arrive at 1 the violet,
and then very quickly at the blue and green, which
together with that violet compound blue, and then at
the yellow and red, by whofe further addition that
blue is converted into whitenefs, which whitenefs con¬
tinues during the tranfit of the edge of the Ruler from
1 to 5, and after that by the fucceffive deficience of
its component Colours, turns firft to compound yellow,
and- then to red: and 1 laft of all the red ceafeth at L.
Then begin the Colours of the fecund Series, which
fucceed in order during the tranfit of the edge of the
Ruler from 5 to O, and are more lively than before,
becaufe more expanded and fevered. And for the
fame reafon, inftead of the former white there inter¬
cedes between the blue and yellow a mixture of orange,
yellow, green, blue and indico, all which together ought
to exhibit a dilute and imperfed green. So the Co¬
lours of the third Series all fucceed in order ; firft, the
violet, which a little interferes with the red of the fe~
cond order, and is thereby inclined to a reddiih purple;
then the blue and green, which are lefs mixed with
other Colours, and confequently more lively than be¬
fore, efpecially the green: Then follows the yellow,
fomeof which towards the green is diftind and good, but
that part of it towards the fucceeding red, as alfo that
red is mixed with the violet and blue of the fourth Se-

ries, whereby Various degrees of red very much incli¬
ning to purple are ’Compounded* This violet and blue,
which fhould fucceed this red, being mixed with, and
hidden in it, there fucceeds a green. And this at firft
is much inclined to blue, but foon becomes a good
green, the only unmixed and lively Colour in this
fourth Series. For as it verges towards the yellow, it
begins to interfere with the Colours of the fifth Series,
by whofe mixture the fucceeding yellow and red are
very much diluted and made dirty, efpecially the yel¬
low, which being the weaker Colour is fcarce able to
(hew it felf. After this the feveral Series interfere more
and more, and their Colours become more and more
intermixed, till after three or four more revolutions
(in which the red and blue predominate by turns )
all forts of Colours are in all places pretty equally ben¬
ded, and compound an even whitenefs.

And fince by the 15th Obfervation the rays indued
with one Colour are tranfmitted, where thofe of ano¬
ther Colour are reflected, the reafon of the Colours
made by the tranfmitted Light in the 9th and 29th Ob-
fervations is from hence evident. ■ ,

If not only the order and fpecies of thefe Colours^
but alfo the precife thicknefs of the plate, or thin body
at which they are exhibited, be defired in parts ;of an
Inch, that may be alfo obtained by affiftance of the 6th
or 16th Obfervations, For according to thofe Obferva-
tions the thicknefs of the thinned Air, which between
two Glades exhibited the molt luminous parts of the
firft fix Rings were ^ ^ gfe parts of

an Inch. Suppofe the Light reflected moft copioufiy
at thefe thickneffes be the bright citrine yellow, or con-

F f 2 fine

[3*]

fine of yellow and orange, and thefe thickneffes will
be G Gv, G ?, G», G i. And this being known, it is
to

h
•

But farther, fincebythe 10th Obfervation the thick-
nefs of Air was to the thicknefs of Water, which be¬
tween the fame Glaffes exhibited the fame Colour, as
4 to 3, and by the a 1 th Obfervation the Colours of
thin bodies are not varied by varying the ambient me¬
dium ; the thicknefs of a Bubble of Water, exhibiting
any Colour, will be ^ of the thicknefs of Air producing
the fame Colour. And fo according to the fame 10th
and aith Obfervations the thicknefs of a plate of
Glafs, whofe refraftion of the mean refrangible ray, is
meafured by the proportion of the Sines 31 to 20,
may be of the thicknefs of Air producing the fame
Colours y and the like of other mediums. I do not
affirm, that this proportion of no to 31, holds in all
the rays; for the Sines of other forts of rays have other
proportions. But the differences of thofe proportions
are fo little that I do not here confider them. On
thefe Grounds I have compofed the following Table^
wherein the thicknefs of Air, Water, and Glafs, at
which each Colour is moft intenfe and fpecifick, is ex-
preffed in parts of an Inch divided into Ten hundred
thoufand equal parts.

etermine what thicknels or Air is reprelented
r by any other diftance of the ruler from

The

‘The ibicknefs of coloured Plates and ‘Particles of

i 'rAea

*®9

hr ft Order,

r Very Black
Black

Beginning of

Their Colours of the j Bll f e kck

White
Yellow-
Orange
: vRed

r Violet
Indica.

Blue

Of the fecond Order, , Q v *:? n

7 < Yellow

■r

Orange
Bright Red
Scarlet

Air, Water . GUfs .

Indico
Blue

Of the third Order, 4 Green

Yellow
Red

L Bluifh Red
( Bluilh Green

Of the fourth Order, jGreen

YYellowiih Green

CRed

Of the fifth Order, S§reenilh Blue

Of the fixth Order, S G reenilh Blue •

^Red

Of the feventh Order, 5 £reeniilv Blue

’ (Ruddy Whitt

t O

3 1

2 O

3 1

i x

1*0

3.5

<>\

1 1 6

12 g

ioi

i 6 f

I 2 f

iof

1-7 9

iif

i Sf

i9f

I 2 f .

1 3 11

22 ri

16 *

2-3 f

J 7ii

157 !

2 51

i 8 *

1 0

I 6 f

2Df

iyi

214

1 si

3 2

20f

25 I

351

2 6 f

22 f

3 6

2.7

23t

4°3

3°i

34l

' 29 I

39i

34'

^n\

4>|w

481

42 .

7 1

53-1

45?

57l

49? . _

■Now if this Table be compared with the 6th Scheme*
you will there fee the conliitution of each Colour* as
to its Ingredients* or the original Colours of which it
is compounded* and thence be enabled to judge of its
intenfenefs or imperfection; which may fuffice in ex¬
plication of the 4-th and 18th Obfervations* unlefs it
be further deiired to delineate the manner how the Co¬
lours appear* when the two ObjeCt-Glaffes are lay’d
upon one another. To do which* let there be de-
fcribed a large Arc of a Circle* and a ftreight Line
which may touch that Arc* and parallel to that Tan¬
gent feveral occult Lines* at fuch distances from it* as
the numbers fet againft the feveral Colours in the Table
denote. For the Arc* and its Tangent* will reprelent
the fuperficies of the Glafles terminating the interjacent
Air; and the places where the occult Lines cut the
Arc will fhow at what diftances from the Center* or
Point of contaft* each Colour is reflected.

There are alfo other ufes of this Table : For by its
afliftanee the thicknefs of the Bubble in the 19th 01>
fervation was determined by the Colours which it ex¬
hibited. And fo the bignefs of the parts of natural
Bodies may be conjectured by their Colours* asfhall be
hereafter fhewn. Alfo* if two or more very thin plates
be lay’d one upon another* fo as to compofe one plate
equalling them all in thicknefs* the refulting Colour
may be hereby determined. For inftance* Mr. Hook in
his Mifcrografhm obferves* that a faint yellow- platemf
Mufcovy-glafs lay’d upon a blue one* conftituted a very
deep purple. The yellow of the firft Order is a faint
one* and the thicknefs of the plate exhibiting it* ac¬
cording to the Table is 4f, to which add 9* the thick-

nefs exhibiting blue of the fecond Order, and the furn
will be which is the thicknefs exhibiting the

purple of the third Order.

To explain, in the next place, the Circumftances of
the ad and 3d Obfervations; that is, how the Rings of
the Colours may (by turning the Prifms about their
common Axis the contrary way to that expreffed in
thofe Obfervations) be converted into white and black
Rings, and afterwards into Rings of Colours again, the
Colours of each Ring lying now in an inverted order; it
muft be remembred, that thofe Rings of Colours are di¬
lated by the obliquaticn of the rays to the Air which
intercedes the Glaffes, and that according to the Table
in the 7th Obfervation, their dilatation or increafe of
their Diameter is moft manifeft and fpeedy when they
are obliqueft, Now the rays of yellow being more re-
frafted by the firft fiiperficies of thefaid Air than thofe
of red, are thereby made more oblique to the fecond fu-
perficies, at which they are reflected to produce the co¬
loured Rings, and confequently the yellow Circle in each
Ring will be more dilated than the red; and the excefs of
its dilatation will be fo much the greater, by how much
the greater is the obliquity of the rays, until at laft it be¬
come of equal extent with the red of the fame Ring. And
for the fame reafon the green, blue and violet, will be alia
fo much dilated by the ftill greater obliquity of their
rays, as to become all very nearly of equal extent with
the red, that is, equally diftant from the center of the
Rings. And then all the Colours of the fame Ring
mud be coincident, and by their mixture exhibit a
white Ring. Arid thefe white Rings muft have black
and dark Rings between them, beeaufe they do not

[ 4° 3

fpread and interfere with one another as before. And
for that reafon alio they muft become diftin&er and vi~
fible to far greater Numbers. But yet the violet being
obliqueft will be fomething more dilated in proportion
to its extent then the other Colours, and fo very apt to
appear at the exterior verges of the white.

Afterwards, by a greater obliquity of the rays, the
violet and blue become more fenfibly dilated than the
red and yellow, and fo being further removed from the
center or the Rings, the Colours muft emerge out of the
white in an order contrary to that which they had be¬
fore, the violet and blue at the exterior limbs of each
Ring,and the red and yellow at the interior. And the vio¬
let, by reafon of the greateft obliquity of its rays, being
in proportion moft of all expanded, will fooneft appear
at the exterior limb of each white Ring, and become
more confpicuous than the reft. And the feveral Series
of Colours belonging to the feveral Rings, will, by
their unfolding and fpreading, begin again to interfere,
and thereby render the Rings lefs diftindf, and not vifL
ble to fo great numbers.

If inftead of the Prifms the Obje£Tglafies be made
ufe of, the Rings which they exhibit become not white
and diftinft by the obliquity of the Eye, by reafon that
the rays in their paflage through that Air which inter¬
cedes the Glaffes are very nearly parallel to thofe Lines
in which they were fir ft incident on the Glaffes, and con-
fequently the rays indued with feveral Colours are not
inclined one more than another to that Air, as it hap¬
pens in the Prifms.

There is yet another circumftance of thefe Experiments
tp be confidered, and that is why the black and white

Rings

[40

Rings which when viewed at adiftance appear diftind,
fhould not only become confufed by viewing them near
at hand , but alfo yield a violet Colour at both the
edges of every white Ring. And the reafon is, that the
rays which enter the Eye at feveral parts of the Pupil,
have feveral obliquities to the Glaffes, and thofe which
are moft oblique, if confidered apart, would reprefent
the Rings bigger than thofe which are the leaft oblique.
Whence the breadth of the perimeter of every white
Ring is expanded outwards by the obliqued rays,
and inwards by the leaft oblique. And this expanfion
is fo much the greater by how much the greater is the
difference of the obliquity ; that is, by how much the
Pupil is wider, or the Eye nearer to the Glaffes. And
the breadth of the violet mu ft be moft expanded, be-
caufe the rays apt to excite a fenfation of that Colour
are moft oblique to a fecond, or further fuperficies of
the thin’d Air at which they are refle&ed, and have
alfo the greateft variation of obliquity, which makes
that Colour fooneft emerge out of the edges of the
white. And as the breadth of every Ring is thus aug¬
mented, the dark intervals muft be diminifhed, until
the neighbouring Rings become continuous, and are
blended, the exterior firft, and then thofe nearer the
Center, fo that they can no longer be diftinguilh’d
apart, but feem to conftitute an even and uniform
whitenel's.

Among all the Obfervations there is none accompa¬
nied with fo odd circumftances as the 24th. Of thofe
the principal are, that in thin plates, which to the
naked Eye ieem of an even and uniform tranfparent

G g white-

r +* ;i

wtiitfcnefs, without any terminations of fhadows, the
refra&ion of a Prifm fhould make Rings of Colours ap¬
pear, whereas it ufually makes ObjeCts appear coloured
only there where they are terminated with fhadows, or
have parts unequally luminous; and that it fhould make
thofe Rings exceedingly diftinCt and white, although
it ufually renders ObjeCts confided and coloured. The
caufe of thefe things you will underhand by considering,
that all the Rings of Colours are really in the plate,
when viewed with the naked Eye, although by reafon
of the great breadth of their circumferences they fo
much interfere and are blended together,that they feem
to conftitute an even whitenefs. But when the rays
pafs through the Prifm to the Eye, the orbits of the
Several Colours in every Ring are refracted, fome more
than others, according to their degrees of refrangibility :
By which means the Colours on one fide of the Ring
(that is on one fide of its Center) become more unfolded
and dilated, and thofe on the other, fide more compli¬
cated and contracted. And where by a due refraCtion
they are fo much contracted, that the fevral Rings be¬
come narrower than to interfere with one another, they
mu ft appear diftinCt, and alfo white, if the conftituent
Colours be fo much contracted as to be wholly coincident.
But, on the other fide, where the orbit of every Ring
is made broader by the further unfolding of its Co¬
lours, it mu ft interfere more with other Rings than
before, and fo become lefs diftinCt.

To explain this a little further, fuppofe the concern
°pig e 7. trick Circles A V, and BX, reprefent the red and violet
of any order, which, together with the intermediate

Colours,

[ 431

Colours, conftitute any one of thefe Rings. Now thefe
being viewed through a Prifm, the violet Circle B X,
will by a greater refraction be further tranflated from
its place than the red A V, and fo approach nearer to
it on that fide, towards which the refractions are made.
For inftance, if the red be tranflated to av, the viole;t
may be tranflated to b x , fo as to approach nearer to it
at x than before, and if the red be further tranflated
to a v, the violet may be fo much further tranflated to
b x as to convene with it at x, and if the red be yet
further tranflated to «■ T , the violet may be ftill fo much
further tranflated to 0 5 as to pals beyond it at 5 , and
convene with it at e and/. And this being underftood
not only of the red and violet, but of all the other in¬
termediate Colours, and alfo of every revolution of
thofe Colours, you will eafily perceive how thofe of the
fame revolution or order, by their nearnefs at xv and
T ?, and their coincidence at xv, e and/, ought to con¬
ftitute pretty diftinCt Arcs of Circles, efpecially at xv,
or at e and /, and that they will appear feverally at
x and at x v exhibit whitenefs by their coincidence,
and again appear feveral at T 5 , but yet in a contrary
order to that which they had before, and ftill retain
beyond and /. But, on the other fide, at ab, ab,
or * 0, thefe Colours mu If become much more confu¬
ted by being dilated and fpread fo, as to interfere with
thofe of other Orders. And the fame confufion will
happen at T 5 between e and/, if the refraction be very
great, or the Prifm very diftant from the ObjeCt-Glaffes:
In which cafe no parts of the Rings will be feen, fave
only two little Arcs at e and/, whofe diftance from one

Gg 2 another,

another will be augmented by removing the Prifhr
ftill further from the ObjedbGlaffes: And thefe little
Arcs muft be diftin<fteft and whiteft at their middle, and
at their ends, where they begin to grow confufed they
muft be coloured. And the Colours at one end of
every Arc muft be in a contrary order to thofe at the
other end, by reafon that they crofs in the interme¬
diate white 5 namely their ends, which verge towards
T ?, will be red and yellow on that fide next the Cen¬
ter, and blue and violet on the other fide. But their
other ends which verge from T ? will on the contrary
be blue and violet on that fide towards the Center, and
on the other fide red and yellow.

Now as all thefe things follow from the Properties
of Light by a mathematical way of reafoning, lo the
truth of them may be manifefted by Experiments. For
in a dark room, by viewing thefe Rings through as*
Prifm, by reflexion of the feveral prifmatique Colours,
which an afliftant caufes to move to and fro upon a
Wall or Paper from whence they are reflected, whilft
the Spectator’s Eye, the Prifm and the ObjeCl-Glafles
( as in the 13 th Obfervation ) are placed fteddy : the
pofition of the Circles made fucceflively by the feveral
Colours, will be found fuch, in relpedt of one another,
as I have defcribed in the Figures or abxv,

or «£ % T. And by the fame method the truth of
the Explications of other Obfervations may be exa¬
mined.

By what hath been faid the like Ph&nomina of
Water, and thin plates of Glafs may be underftoodi
But in fmall fragments of thofe plates, there is this

further

further oblervable, that where they lye flat upon a
Table and are turned about their Centers whilft they are
viewed through a Prifm, they will in Tome poftures
exhibit waves of various Colours, and fome of them ex¬
hibit thefe waves in one or two pofitions only, but the
moft of them do in all pofitions exhibit them, and make
them for the moft part appear almoft all over the plates.
The reafon is, that the fuperficies of fuch plates are not
even, but have many cavities and fwellings, which how
shallow foever do a little vary the thicknefs of the
plate. For at the feveral fides of thofe cavities, for
the reafons newly defcribed, there ought to be produ¬
ced waves in feveral poftures of the Prifm. Now though
it be but fome very lmall, and narrower parts of the
Glafs, by which thefe waves for the moft part are cau-
led, yet they may feem to extend themfelves over the
whole Glafs, beeaulefrom the narroweft of thofe parts
there are Colours of feveral Orders that is of feveral
Rings, confufedly reflected, which by refradion of the
Prifm are unfolded, feparated, and according to their
degrees of refradion, difperfed to feveral places, fo as to.
conftitute fo many feveral waves, as there were divers-
orders of Colours promifcuoufly refleded from that
part of the Glafs.

Thefe are the principal Phenomena of thin Plates
or Bubbles, whole explications depend on the pro¬
perties of Light, which I have heretofore delivered;
And thefe you fee do neceflarily follow from them, and
agree with them, even to their very leaft circumftancesj
and not only fo, but do very much tend to their proof
Thus, by the 14th Obfervation, it appears, that the

rays ■

[ 4*1

rays of leveral Colours made as well by thin Plates or
Bubbles, as by refractions of a Prifm, have feveral de¬
grees of refrangibility, whereby thofe of each order,
which at their reflexion from the Plate or Bubble are
intermixed with thofe of other orders, are feparated
from them by refraCtion, and affociated together fo as to
become vifibleby themfelves like Arcs of Circles. For
if the rays were all alike refrangible, ’tis impoffible that
the whitenels, which to the naked fence appears uni¬
form, ffiould by refraction have its parts tranlpofed and
ranged into thofe black and white Arcs.

It appears alfo that the unequal refractions of dif-
form rays proceed not from any contingent irregulari¬
ties ; fuch as are veins, an uneven polilh, or fortuitous
pofition of the pores of Glafs; unequal and cafual mo¬
tions in the Air or iEther; the fpreading, breaking, or
dividing the fame ray into many diverging parts, or
the like. For, admitting any fuch irregularities, it would
be impoffible for refractions to render thofe Rings fo
very diftinCt, and well defined, as they do in the
aqth Obfervation. It is neceflary therefore that eve¬
ry ray have its proper and conftant degree of refran¬
gibility connate with it,according to which its refraCtion
is ever juftly and regularly performed, and that feve¬
ral rays have feveral of thofe degrees.

And what is faid of their refrangibility may be alfo
underftood of their reflexibility, that is of their difpo-
fitions to be reflected fome at a greater, and others at a
lefs thicknels, of thin Plates or Bubbles, namely, that
thofe difpofitions are alfo connate with the rays, and
immutable; as may appear by the 15th, 1 qth, and

15th

[473

t 5th Observations compared with the fourth and
eighth.

By the precedent Observations it appears alfo, that
whitenefs is a diffimilar mixture of all Colours, and that
Light is a mixture of rays indued with all thofe Co¬
lours. For confidering the multitude of the Rings of
Colours, in the 3d, xath and 24th Observations, it is
manifeft that although in the 4th and 18th Observa¬
tions there appear no more than eight or nine of thofe
Rings, yet there are really a far greater number, which.
So much interfere anduningle with one another, as after
thofe eight or nine revolutions to dilute one another
wholly, and constitute an even and fenfibly uniform
whitenefs. And consequently that whitenefs muft be
allowed a mixture of all Colours, and the Light which
conveys it to the Eye muft be a mixture of rays indued
with all thofe Colours.

But further, by the 14th Observation, it appears,
that there is a conftant relation between Colours and
Refrangibility, the moft refrangible rays being violet,
the leaft refrangible red, and thofe of intermediate Co¬
lours having proportionably intermediate degrees of re¬
frangibility. And by the 13 th, 14th and 15th Obser¬
vations, compared with the 4th or 18th, there appears
to be the fame conftant relation between Colour and
Reflexibility, the violet being in like circumftances re-
fleded at leaft thickneflfes of any thin Plate or Bubble,
the red at greateft thickneflfes, and the intermediate
Colours at intermediate thickneflfes. Whence it fol¬
lows, that the colorifique difpoiitions of rays are alfo
connate with them and immutable, and by confequence

that

. [+ 8 ]

that all the productions and appearances of Colours
in the World are derived not from any phyfical change
caufed in Light by refraction or reflexion, but only
from the various mixtures or reparations of rays, by
virtue of their different Refrangibility or Reflexibility.
And in this refpeCt the Science of Colours becomes a
Speculation as truly mathematical as any other part of
Optiques. I mean fo far as they depend on the nature
of Light, and are not produced or altered by the power
of imagination, or by ftriking or prefling the Eyes.

THE

Book,H. . Plate!.

Fig. 2.

[ 49 ]

THE

SECOND BOOK

OPT I c k s.

PART III.

Of the permanent Colours of natural Bodies y and the
+Analogy between them and the Colours of thin tranf*
parent Blates .

I Am now come to another part of this Defign, which
is to conlider how the Phenomena of thin tranfpa-
rent Plates hand related to thol e of all other natural
Bodies* Of thefe Bodies I have already told you that
they appear of. divers Colours, accordingly as they are
difpofed to reflect moft copioufly the rays originally
indued with thofe Colours* But their Conftitutions f
whereby they reflect fome rays more copioufly than
others, remains to be difcovered, and thefe I fliall en*
deavour to manifeft in the following Propofi.tio.ns,

Hh PROP*

t $0 ]

PROP, I.

T’hoft b fiiperficiesoftr an [parent Bodies refieSl the gr eat eff
quantify of Light ^ ‘which have the great eft refrailing powers
that is , which intercede mediums that differ moft in their
refractive denfities* And in the confines of equally re-
frailing mediums there is no reflexion .

The Analogy between reflexion and refraction will
appear by confidering, that when Light paffeth ob¬
liquely out of one medium into another which refraCts
from the perpendicular, the greater is difference of
their refractive denfity, the lefs obliquity is requifite
to caufea total reflexion. For as the Sines are which
meafure the refraction, fo is the Sine of incidence at
which the total reflexion begins, to the radius of the
Circle, and eonfequently that incidence is leaf! where
there is the greateft difference of the Sines. Thus in the
paffing of Light out of Water into Air, where the
refraCtion is mealured by the Ratio of the Sines 3 to 4,
the total reflexion begins when the Angle of incidence
is about 48 degrees 3 5 minutes. In paffing out ofGlafs
into Air, where the refraCtion is mealiired by the Ratio
of the Sines 20 to 31, the total reflexion begins when
the Angle of incidence is 40 deg. 10 min. and fo in
paffing out of cryftal, or more ftrongly refracting me¬
diums into Air, there is ftill a lefs obliquity requifite
to caufe a total reflexion. Superficies therefore which
refraCt moft do fooneft reflect all the Light which is in¬
cident on them, and fo mnft be allowed moft ftrongly
reflexive*

But

[*<] .

But the truth of this Propofition wilt further appear

by obferving, that in the fuperficies interceding two
tranfparent mediums, fuch as are (Air,Water,Oyl,Com¬
mon-Glafs, Cryftal, Metalline-Glafles, Ifland-Glafles,
white tranfparent Arfnick, Diamonds, I 2V. ) the re¬
flexion is ftronger or weaker accordingly, as the fuper-
ficies hath a greater or lefs refracting power. For in
the confine of Air and Sal-gemm 'tis ftronger than in
the confine of Air and Water, and Hill ftronger in the
confine of Air and Common-Glafs or Cryftal,and ftronger
in the- confine of Air and a Diamond. If any of thefe,and
fuch like tranfparent Solids, be immerged in Water, its
reflexion becomes much weaker than before, and ftill
weaker if they be immerged in the more ftrongly re¬
fracting Liquors of well-reCtified oyl of Vitriol or fpirit
of Turpentine. If Water be diftinguifhed into two parts,
by any imaginary furface, the reflexion in the confine
of thofe two parts is none at all. In the confine of Wa¬
ter and Ice ’tis very little, in that of Water and Oyl ’tis
fomething greater, in that of Water and Sal-gemm ftill
greater, and in that of Water and Glafs, or Cryftal, or
other denfer lubftances ftill greater, accordingly as thofe
mediums differ more or lefs in their refracting powers.
Hence in the confine of Common-Glafs and Cryftal,
there ought to be a weak reflexion, and a ftronger re¬
flexion in the confine of Common and Metalline-Glafs,
though I have not yet tried this. But, in the confine of
two Glafles of equal denfity, there is not any lenfible re¬
flexion, as was Ihewn in the firft Obfervation. And
the fame may be underftood of the fuperficies interce¬
ding two Cryftals, or two Liquors, or any other Sub-
ftances in which no refraction is caufed. So then the
' • Hh 2 sreafon

reafoii why uniform pellucid mediums, (fuch as Water,.
Glafs, or Cryftal) have no fenfible reflexion but in
their external fuperficies, where they are adjacent to
other mediums of a different denfity, is becaufe all
their contiguous parts have one and the fame degree
of denfity,

PROP, IL

The leafl farts of almoji all natural Bodies are in fome
meafure tranffarent : And the opacity of tbofe Bodies
arifeth from the multitude of reflexions caufed in their in*
ternal Barts .

That this is fo has been obferved by others, and-
will eafily be granted by them that have been conver-
fant with Mifcrofcopes. And it may be alfo tryed by
applying any fubftance to a Hole through which fome
Light is immitted into a dark room. For how opake
foever that fubftance may feem in the open Air, it will
by that means appear very manifeftly tranfparent, if
it be of a fufficient thinnefs. Only white metalline Bo¬
dies muft be excepted, which by reafon of their excef-
five denfity feem to reflect almoft all the Light inci¬
dent on their firft fuperficies , unlefs by folution in
menftruums they be reduced into very fmall particles^
■and then they become tranfparent*

PROP* 1 IL

Between the parts of opake and coloured Bodies are
many ffacesj either empty or replenijhed , with mediums
of other densities ; as Water between the tinging corpufcles
wherewith my Liquor is impregnated y Air between the

aqueom

aqueous globules that conjlitute Clouds or Mtfts j and for
the mofl fart [faces void of both .Air and TVat er^ but yet
ferhafs not wholly void of all fubjlance , between the farts
of hard Bodies ,

The truth of this is evinced by the two precedent
Propofitions : For by the fecond Proportion there are
many reflexions made by the internal parts of Bodies^
which, by the firft Propofition, would not happen if
the parts of thofe Bodies were continued without any
luch interftices between them, becaufe reflexions are
caufed only in fuperficies, which intercede mediums of
a differing denfity by Prop, r.

But further, that this difcontinuity of parts is the -
principal caufe of the opacity of Bodies,, will appear by
confidering, that opake fubftances become tranfparent
by filling their pores with any fubftance of equal or al-
mofl: equal denfity with their parts. Thus Paper dip¬
ped in Water or Oyl, the Qculm mundi Stone fteep’d in
Water, Linnen-cloth oyled or varniflied, and many other
fubftances foaked in fuch Liquors as will intimately
pervade their little pores, become by that means more
tranfparent than otherwise ; fo, on the contrary, the.
mofl: tranfparent fubftances may by evacuating their
pores, or Separating their parts, be rendred fuffieiently
opake, as Salts or wet Paper, or the 0 culm mundi Stone
by being dried, Horn by being fcraped, Glafs by being
reduced to powder, or other wife flawed^ Turpen¬
tine by being ftirred about with Water till they mix,
imperfectly, and Water by being formed into many
fmall Bubbles, either alone in the form of froth, or
by fhaking it together with Oyl of Turpentineor
with fome other convenient Liquor, with which if will-

not

C54]

not perfectly incorporate. And to the increafe of the
opacity of thefe Bodies it conduces fomething, that by
the 1 3 th Obfervation the reflexions of very thin tranf-
parent fubftances are confiderably ftronger than thofe
made by the fame fubftances of a greater thicknefs.

PROP. IV.

'1 he -parts of Bodies and their Interfiles muft not be
lefs than, of fame definite bignefs , to render them opake and
coloured.

For the opakeft Bodies, if their parts be fubtily
divided, (as Metals by being diflolved in acid men-
ftruums, ®c.) become perfectly tranfparent. And you
may alfo remember, that in the eighth Obfervation
there was no fenfible reflexion at the fuperficies of
the Object-GlaiTes where they were very near one
another, though they did ndt abfolutely touch. And
in the r yth Obfervation the reflexion of theWater-bubble
where it became thinneft was almoft infenfible, fo as
to caufe very black Spots to appear on the top of the
Bubble by the want of reflected Light.

On thefe grounds I perceive it is that Water, Salt,
Glals, Stones, anclfuch like fubftances, are tranfparent.
For, upon divers confiderations, they feem to be as full
of pores or interftiees between their parts as other Bo¬
dies are, but yet their parts and interftiees to be too
fmall to caufe reflexions in their common furfaces.

PROP.

[ 55 ]

PROP. V.

The tranfparent parts of Bodies according to their fe*
veral fizes mufl re fled rays of one Colour* and tranfmit
thofe of another , on the fame grounds that thin 2 ?lattes or
Bubbles do reflect or tranfmit thofe rays . Tlnd this I take
to be the ground of all their Colours.

For if a thiif d or plated Body, which being of an
even thicknefs, appears all over of one uniform Co¬
lour, fhould be flit into threds, or broken ■ into frag¬
ments, of the fame thicknefs with the plate; I fee no
reafon why every thred or fragment fhould not keep its
Colour, and by confequence why a heap of thofe threds
or fragments fhould not conftitute a mafs or powder of.
the fame Colour, which the plate exhibited before it
was broken. And the parts of all natural Bodies being
like fo many fragments of a Plate, muft on the fame
grounds exhibit the fame Colours.

Now that they do fo, will appear by the affinity of
their properties. The finely coloured Feathers of fomer
Birds, and particularly thofe of Peacocks Tails, do im
the very fame part of the Feather appear of feveral Co¬
lours in feveral politicos of the Eye, after the very fame
manner that thin Plates were found to do in the yth:
and 19th Obfervations, and therefore arife from the
thinnefs of the tranfparent parts of the Feathers y that
is, from the flendernefs of the very fine Hairs, or Capitt
mentaj which grow out of the tides of the gruffer late¬
ral, branches or fibres of thofe Feathers,. And;to,the
fame purpofe it is, that the Webs of. feme Spiders- by:

, _ ~. ~ Wng :

being fpun very fine have appeared coloured, as forne
have obferved, and that the coloured fibres of fome lilks
by varying the pofition of the Eye do vary their Co¬
lour. Alfo the Colours of filks, cloths, and other fub-
ftances, which Water or Oyl can intimately penetrate,
become more faint and obfcure by being immerged in
thofe liquors, and recover their vigor again by being
dried, much after the manner declared of thin Bodies
in the ioth and aith Obfervations. Leaf-gold, fome
forts of painted Glafs, the infufion of Lignum Nefbn-
ticum , and fome other fubftances reflect one Colour,
and tranfinit another, like thin Bodies in the 9th and
aoth Obfervations. And fome of thofe coloured pow¬
ders which Painters ufe, may have their Colours a little
changed, by being very elaborately and finely ground.
Where I fee not what can be juftly pretended for thofe
changes, befides the breaking of their parts into lefs
parts by that contrition after the fame manner that the
Colour of a thin Plate is changed by varying its thick-
nefs. For which reafon alfo it is that the coloured flowers
of Plants and Vegitables by being bruifed ufually be¬
come more tranfparent than before, or at leaft in fome
degree or other change their Colours. Nor is it much
lefs to my purpofe, that by mixing divers liquors very-
odd and remarquable productions and changes of Co¬
lours may be effected, of which no caufe can be more
obvious and rational than that the faline corpufcles of
one liquor do varioufly a£t upon or unite with the
tinging corpufcles of another, fo as to make them fwell,
or Ihrink (whereby not only their bulk but their den-
fity alfo may be changed ) or to divide them into
fmaller corpufcles, (whereby a coloured liquor may be¬
come

come tranfparent) or to make many of them affociate
into one duller, whereby two tranfparent liquors may
compofe a coloured one. For we fee how apt thofe
faline menftruums are to penetrate and diffolve fub-
ftances to which they are applied, and fome of them
to precipitate what others diffolve. In like manner, if
we confider the various Phenomena of the Atmofphaere,
we may obferve, that when Vapors are firft raifed, they
hinder not the tranfparency of the Air, being divided
into parts too fmall to caui’e any reflexion in their fuper-
ficies. But when in order to compofe drops of rain they
begin to coalefce and conftitute globules of all inter¬
mediate fizes, thole globules when they become of a
convenient fize to relied! fome Colours and tranfmit
others, may conftitute Clouds of various Colours accor¬
ding to their fizes. And I fee not what can be ratio¬
nally conceived in fo tranfparent a fubftance as Water for
the production of thefe Colours, befides the various
fizes of its fluid and globuler parcels.

PROP. VI.

The farts of Bodies on which their Colours defend
are denfer than the medium , which -pervades their m-
terftices.

This will appear by confidering, that the Colour of
a Body depends not only on the rays which are inci¬
dent perpendicularly on its parts, but on thofe alfo
which are incident at all other Angles. And that ac¬
cording to the 7th Obfervation, a very little variation
of obliquity will change the 1'efledted Colour where the
thin body or fmall particle is rarer than the ambient

I i medium,

medium, infomuch that fuch a fmall particle will tit di-
verily oblique incidences reflect all forts of Colours, in
fo great a variety that the Colour refulting from them
all, confufedly reflected from a heap of fuch particles,
muft rather be a white or grey than any other Colour,
or at beft it muft be but a very imperfect and dirty Co¬
lour. Whereas if the thin body or fmall particle be
much denfer than the ambient medium, the Colours
according to the 19th Obfervation are fo little changed
by the variation of obliquity, that the rays which are
reflected leaft obliquely may predominate over the reft
fo much as to caufe a heap of fuch particles to appear
very intently of their Colour.

It conduces alfo fomething to the confirmation of this
Propolition, that, according to the 22th Obfervation,
the Colours exhibited by the denfer thin body within
the rarer, are more brifque than thofe exhibited by the
rarer within the denfer.

PROP. VII.

The big nefs of the component parts of natural Bodies
may be conjectured by their Colours .

For fince the parts of thefe Bodies by Prop. 5. do
moft probably exhibit the fame Colours with a Plate of
equal thicknefs^ provided they have the fame refraftive
denfity; and fince their parts Teem for the moft part to
have much the fame denfity with Water or Glafs, as
by many circumftances is obvious to colle£t; to deter¬
mine the fizes^of thofe parts you need only have recourfe
to the precedent Tables, in which the thicknefs of Wa¬
ter or Glals exhibiting any Colour is expreffed. Thus

[59-].

if it be defired to know the Diameter of a corpufcle,
which being of equal denfity with Glafs fhall refled
green of the third order; the number 16- lhews it to
be l6 ^ parts of an Inch.

IOOOOO A

The greateft difficulty is here to know of what order
the Colour of any Body is. And for this end we muft
have recourfe to the 4th and 18th Obfervations, from
whence may be collected thefe particulars.

Scarlets , and other reds , oranges and yellows^ if they
be pure and intenfe are moft probably of the fecond or¬
der. Thofe of the firft and third order alfo may be
pretty good, only the yellow of the firft order is faint,
and the orange and red of the third order have a great
mixture of violet and blue.

There may be good greens of the fourth order, but
the pureft are of the third. And of this order the green
of all vegitables feem to be, partly by reafon of the in-
tenfenefs of their. Colours, and partly becaufe when
they wither fome of them turn to a greenilh yellow,
and others to a more perfect yellow or orange, or per¬
haps to red, paffing firft through all the aforefaid in¬
termediate Colours. Which changes feem to be effe&ed
by the exhaling of the moifture which may leave the
tinging corpufcles more denfe, and fomething augmen¬
ted by the accretion of the oyly and earthy part of
that moifture. Now the green without doubt is of the
fame order with thofe Colours into which it changeth,
becaufe the changes are gradual, and thofe Colours,
though ulually not very full, yet are often too full and
lively to be of the fourth order.

li a

Blues

[do]

Blues and purples maybe either of the fecond or third
order, but the beft are of the third. Thus the Colour
of violets feems to be of that order, becaufe their Syrup
by acid Liquors turns red, and by urinous and alcali-
zale turns green. For fince it is of the nature of Acids
to diffolve or attenuate, and of Alcalies to precipitate
or incraflate, if the purple Colour of the Syrup was of
the fecond order, an acid Liquor by attenuating its ting¬
ing corpufcles would change it to a red of the fir ft
order, and an Alcaly by incrafifating them would change
it to a green of the fecond order 3 which red and green,
efpecially the green, feem too imperfect to be the Co¬
lours produced by thefe changes. But if the faid purple
be fuppofed of the third order, its change to red of the
fecond, and green of the third, may without any in¬
convenience be allowed.

If there be found any Body of a deeper and lei's red-
difh purple than that of the violets, its Colour moft
probably is of the fecond order. But yet their being
no Body commonly known whofe Colour is conftantly
more deep than theirs, 1 have made ufe of their name to
denote the deepeft and leaft reddifih purples, fuch as
manifeftly tranfcend their Colour in purity.

The Hue of the firft order, though very faint and
little, may poffibly be the Colour of fome fubftances;
and particularly the azure Colour of the Skys feems to
be of this order. For all vapours when they begin to
condenfe and coalefce into lmall parcels, become firft of
that bignefs whereby fuch an Azure muft be reflected
before they can conftitute Clouds of other Colours. And
fo this being the firft Colour which vapors begin to
reflect, it ought to be the Colour of the hneft and moft <

tranfe

[ 6 !]

tranfparent Skys in which vapors are not arrived to that
grofnefs requifite to reflect other Colours, as we find it
is by experience.

JVbiteneJ's , if moil: intenfe and luminous, is that of the
firft order, if lefs ftrong and luminous a mixture of the
Colours of feveral orders. Of this laft kind is the
whitenefs of Froth, Paper, Linnen, and moft white fub-
fiances ; of the former I reckon that of white metals to
be. For whilft the denfeft of metals, Gold, if foliated
is tranfparent, and all metals become tranfparent if
diffolved in menftruums or vitrified, the opacity of
white metals arileth not from their denfity alone. They
being lefs denfe than Gold would be more tranfparent
than it, did not feme other caufie concur with their den¬
fity to make them opake. And this caufe I take to be
fuch a bignefs of their particles as fits them to reflect
the white of the firft order. For if they be of other
thickneffes they may reflect other Colours, as is mani-
left by the Colours which appear upon hot Steel in tem¬
pering it, and fometimes upon the ferface of melted
metals in the Skin or Scoria which arifes upon them in
their cooling. And as the white of the firft order is
the ftrongeft which can be made by Plates of tranfparent
lubftances, fo it ought to be ftronger in the denfer fub~
fiances of metals than in the rarer of Air, Water and
Glafs. Nor do I fee but that metallic lubftances of fuch
a thicknefs as may fit them to refleft the white of the
firft order, may, by reafon of their great denfity (accor¬
ding to the tenour of the firft of thefe Propofitions) re¬
flect all the Light incident upon them, and fo be as
opake and fplendent as its poffible for any Body to be.
Gold, or Copper mixed with lefs than half their weight

of Silver, or Tin, or Regulus of Antimony, in fufion
or amalgamed with a very little Mercury become white;
which Thews both that the particles of white metals
have much more fuperficies, and fo are fmaller, than
thofe of Gold and Copper, and alfo that they are lb
opake as not to fuffer the particles of Gold or Copper to
Ihine through them. Now it is fcarce to be doubted,
but that the Colours of Gold and Copper are of the fe-
cond or third order, and therefore the particles of white
metals cannot be much bigger than is requifite to make
them refled, the white of the firft order. The volati¬
lity of Mercury argues that they are not much bigger,
nor may they be much lefs, leaft they lofe their opacity,
and become either tranfparent as they do when attenua¬
ted by vitrification, or by folution in menftruums, or
black as they do when ground fmaller, by rubbing Sil¬
ver,or Tin, or Lead, upon other fubftances to draw black
Lines. The firft and only Colour which white metals
take by grinding their particles fmaller is black, and
therefore their white ought to be that which borders
upon the black Spot in the center of the Rings of Co¬
lours, that is, the white of the firft order. But if you
would hence gather the bignefs of metallic particles,
you muft allow for their denfity. For were Mercury
tranfparent, its denfity is fuch that the Sine of inci¬
dence upon it (by my computation) would be to the
line of its refraction, as 71 to 20, or 7 to 2. And
therefore the thicknefs of its particles, that they may
exhibit the fame Colours with thofe of Bubbles of Wa¬
ter, ought to be lefs than the thicknefs of the Skin of
thofe Bubbles in the proportion of 2 to 7. Whence
its poffible that the particles of Mercury may be as little

, 1

I f? I

as the particles of lome tranfparent and volatile fluids,
and yet reflect the white of the firft order.

Laftly, for the production of blacky the corpufcles
muft be lefs than any of thofe which exhibit Colours.
For at all greater fizes there is too much Light refle¬
cted to conftitute this Colour. But if they be fuppo-
fed a little lefs than is requifite to refieCt the white and
very faint blue of the firft order, they will, according
to the 4th, 8th, 17th and 18th Oblervations, refleCt
fo very little as to appear intenfly black, and yet may
perhaps varioully refraCt it to and fro within them*
felves fo long, until it happen to be ftifled and loft,
by which means they will appear black in all pofitions
of the Eye without any tranfparency. And from hence
may be underftood why Fire, and the more fubtile
difiblver Putrefaction, by dividing the particles of tub-
fiances, turn them to black, why fmall quantities of
black fubftances impart their Colour very freely and in-
tenfly to other fubftances to which they are applied •
the minute particles of thefe, by reafon of their very
great number, eafily overfpreading the grofs particles
of others; why Glafs ground very elaborately with
Sand on a copper Plate, ’till it be well polifhed, makes
the Sand, together with what is worn off from the Glafs
and Copper, become very black : why black fubftances
do fooneft of all others become hot in the Suffs Light
and burn, (which effeCt may proceed partly from the
multitude of refractions in a little room, and partly
from the eafy commotion of fo very fmall corpufcles;)
and why blacks are ufually a little inclined to a bluifh
Colour. For that they are fo may be feen by illumina¬
ting white Paper by Light reflected from black fub¬
ftances.

[ *4 ]

fiances. For the Paper will ufually appear of a biuifh
white; and the reafon is, that black borders on the
obfcure blue of the firft order defcribed in the 18th
Obfervation, and therefore reflects more rays of that
Colour than of any other.

In thefe Defcriptions I have been the more particu¬
lar, becaufe it is not impoffible but that Mifcrofcopes
may at length be improved to the difcovery of the
particles of Bodies on which their Colours depend, if
they are not already in fome meafure arrived to that de¬
gree of perfection. For if thofe Inftruments are or can
be fo far improved as with fufficient diftinCtnefs to re-
prefent Objects five or fix hundred times bigger than
at a Foot diftanoe they appear to our naked Eyes, I
fhould hope that we might be able to difcover foine of
the greateft of thofe corpufcles. And by one that would
magnify three or four thoufand times perhaps they
might all be difcovered, but thofe which produce black-
nefs. In the mean while I fee nothing material in this
Difcourfe that may rationally be doubted of excepting
this Pofition, That traniparent corpufcles of the fame
thicknefs and deniity with a Plate, do exhibit the fame
Colour. And this I would have underfiood not with¬
out fome latitude, as well becaufe thofe corpufcles may
be of irregular Figures, and many rays muft be oblique¬
ly incident on them, and fo have a Ihorter way through
them than the length of their Diameters, as becaufe the
ftraitnefs of the medium pent in on all fides within fuch
corpufcles may a little alter its motions or other qua¬
lities on which the reflexion depends. But yet I can¬
not much lufpeCt the laft, becaufe I have obferved of
dome fimall Plates of Mulcovy-Giafs which were of an

even

even thicknefs, that through a Mifcrofcope they have
appeared of the fame Colour at their edges and cor*
ners where the included medium was terminated, which
they appeared of in other places* However it will add
much to our fatisfa&ion, if thofe corpufcles could be dis¬
covered with Mifcrofcopes; which if we {hall at length
attain to, I fear it will be the utmoft improvement of
this fenfe. For it feems impoffible to fee the more fe~
cret and noble works of nature within the corpufcles
by reafon of their transparency*

PRO P. VIII.

T*he caufe of Reflexion is not the impinging of Light on
the folid or imferviom farts of Bodies , as is commonly he*
lieved .

This will appear by the following Confiderations.
Firft, That in the paffage of Light out of Glafs into
Air there is a reflexion as ftrong as in its paffage out of
Air into Glafs, or rather a little ftronger, and by many
degrees ftronger than in its paffage out of Glafs into
Water. And it feems not probable that Air fliould have
more reflecting parts than Water or Glafs. But if that
fliould poflibly be fuppofed, yet it will avail nothing ;
tor the reflexion is as ftrong or ftronger when the Air is
drawn away from the Glafs, (fuppofe in the Air-pump
invented by Mr. Boyle ) as when it is adjacent to'it.
Secondly, If Light in its paffage out of Glafs into Air
be incident more obliquely than at an Angle of 40 or
41 degrees it is wholly reflected, if lefs obliquely it is
in great meafure tranfmitted. Now it is not to be ima^
gined that Light at one degree of obliquity fliould meet

K k with

[<56]

with pores enough in the Air to tranfmit the greater
part of it, and at another degree of obliquity fihould
meet with nothing but parts to reflefk it wholly, efpe-
dally confidering that in its pafiage out of Air into
Glafs, how oblique foever be its incidence, it finds
pores enough in the Glafs to tranfmit the greateft part
of it. If any Man fuppofe that it is not reflected by the
Air, but by the outmoft fuperficial parts of the Glafs,
there is ftill the fame difficulty : Befides, that fuch a
Suppofition is unintelligible, and will alfo appear to be
falle by applying Water behind fome part of the Glafs
inftead of Air. For fo in a convenient obliquity of the
rays fuppofe of 45 or 46 degrees, at which they are all
reflected where the Air is adjacent to the Glals, they
lhall be in great meafure tranfmitted where the Water
is adjacent to it; which argues, that their reflexion
or tranliniffion depends on the conftitution of the Air
andi Water behind the Glafs, and not on the ftriking
off the rays upon the parts of the Glafs. Thirdly, If
the Colours made by a Prifm placed at the entrance of
a beam of Light into a darkened room be fucceffively
caft bn a fecond Prifm placed at a greater diftance from
the former, in fuch manner that they are all alike inci¬
dent upon it, the fecond Prifin may be fo inclined to
the incident rays, that thofe which are of a blue Colour
lhall be all refleded by it, and yet thofe of a red Colour
pretty copioufly tranfmitted. Now if the reflexion be
caufed by the parts of Air or Glafs, I would ask, why
at the fame obliquity of incidence the blue fliould whol¬
ly impinge on thofe parts fo as to be all reflected, and
yet the red find pores enough to be in great meafure
tranfmitted. Fourthly, where two Glailes touch one

another.

another, there is no fenfible reflexion as was declared
in the firft Obfervation; and yet I fee no reafon why
the rays fhould not impinge on the parts of Glafs as
much when contiguous to other Glafs as when con¬
tiguous to Air. Fifthly, When the top of a Water-
bubble (in the 17th Obfervation) by the continual fub-
fiding and exhaling of the Water grew very thin, there
was fuch a little and almoft infenfible quantity of Light
reflected from it, that it appeared intenlly black ; where¬
as round about that black Spot, where the Water was
thicker, the reflexion was fo ftrong as to make the
Water feem very white. Nor is it only at the leaft
thicknefs of thin Plates or Bubbles, that there is no
manifeft reflexion, but at many other thicknefles con¬
tinually greater and greater. For in the 15 th Obfer¬
vation the rays of the fame Colour were by turns tranf-
mitted at one thicknefs, and relieved at another thick¬
nefs, for an indeterminate number of fucceffions. And
yet in the fiiperficies of the thinned Body, where it is
of any one thicknefs, there are as many parts for the
rays to impinge on, as where it is of any other thick¬
nefs. Sixthly, If reflexion were caufed by the parts of
refle&ing Bodies, it would be impoffible for thin Plates
or Bubbles at the fame place to reflect the rays of one
Colour and tranfmit thofe of another, as they do accor¬
ding to the 13 th and 15 th Obfervations. For it is
not _ to be imagined that at one place the rays which
for inftance exhibit a blue Colour, fhould have the for¬
tune to dafh upon the parts, and thofe which exhibit
a red to hit upon the pores of the Body ; and then at
another place, where the Body is either a little thicker,
or a little thinner, that on the contrary the blue fhould

K k 2 hit

[68]

hit upon its pores., and the red upon its parts* Laftly,
were the rays of Light reflected by impinging on the
folid parts of Bodies, their reflexions from polifhed Bo¬
dies could not be fo regular as they are* For in po-
lilhing Glafs with Sand, Putty or Tripoly, it is not to
be imagined that thofe fubftances can by grating and
fretting the Glafs bring all its leaft particles to an ac¬
curate polilh ; fo that all their furfaces fliall be truly
plain or truly fpherical, and look all the fame way, fo
as together to compofe one even fur face* The fmaller
the particles of thofe fubftances are, the fmaller will
be the fcratches by which they continually fret and wear
away the Glafs until it be polifhed, but be they never
fo fmall they can wear away the Glafs no otherwife
than by grating and fcratching it, and breaking the
proturberances, and therefore polifti it no otherwife
than by bringing its roughnefs to a very fine Grain, fo
that the fcratches and frettings of the furface become
too fmall to be vifible. And therefore if Light were
reflected by impinging upon the folid parts of the Glafs^
it would be fcattered as much by the moft polifhed
Glafs as by the roughed. So then it remains a Pro¬
blem, how Glafs polifhed by fretting fubftances can re¬
flect Light fo regularly as it does. And this Problem
is fcarce otherwife to be folved than by faying, that
the reflexion of a ray is effected, not by a Angle point of
the reflecting Body, but by fome power of the Body
which is evenly diffufed all over its furface, and by
which it aCts upon the ray without immediate contaCt.
For that the parts of Bodies do aCt upon Light at a di-
fiance fliall be fhewn hereafter.

Now

Now if Light be reflected not by impinging on the
folid parts of Bodies, but by fome other principle ; its
probable that as many of its rays as impinge on the
folid parts of Bodies are not reflected but ftifled and
loft in the Bodies. For otherwife we muft allow two
forts of reflexions. Should all the rays be reflected which
impinge on the internal parts of clear Water or Cryftal,
thofe fubftances would rather have a cloudy Colour
than a clear tranfparency. To make Bodies look black,
its neceflary that many rays be ftopt, retained and loft
in them, and it feems not probable that any rays can
be ftopt and ftifled in them which do not impinge on
their parts.

And hence we may underftand that Bodies are much
more rare and porous than is commonly believed. Wa¬
ter is 19 times lighter, and by confequence 19 times
rarer than Gold, and Gold is fo rare as very readily
and without the leaft oppofition to tranfmit the mag-
netick Effluvia,, and eaiily to admit Quick Tilver into
its pores, and to let Water pafs through it. For a con>
cave Sphere of Gold filled with Water, and fodered up,
has upon preffing the Sphere with great force, let the
Water fqueeze through it, and ftand all over its out-
fide in multitudes of fmall Drops, like dew, without
burfting or cracking the Body of the Gold as I have
been informed by an Eye-witnefs. From all which we
may conclude, that Gold has more pores than folid
parts, and by confequence that Water has above forty-
times more pores than parts. And he that ftiall find out
anHypothefis, by which Water may befo rare, and yet
not be capable of compreflion by force, may doubtleis
by the fame Hypothefis make Gold and Water, and all

other

[ 7 ° 3

other Bodies as much rarer as he pleafes, fo that Light
may find a ready paflage through tranfparent fub-

ftances.

PROP. IX.

Bodies reflect and refrafi Light by one and the fame
fower varioujly exercifed in variom circumflances.

This appears by feveral Confiderations. Firft, Be-
caufe when Light goes out of Glafs into Air, as ob¬
liquely as it can poffibly do, if its incidence be made
ftill more oblique, it becomes totally reflected. For
the power of the Glafs after it has refracted the Light
as obliquely as is poffible if the incidence be ftill made
more oblique, becomes too ftrong to let any of its rays
go through, and by confequence caufes total reflexions.
Secondly, Becaufe Light is alternately reflected and
tranfmitted by thin Plates of Glafs for many fucceflions
accordingly, as the thicknefs of the Plate increafes
in an arithmetical Progreffion. For here the thicknefs
of the Glafs determines whether that power by which
Glafs ads upon Light ftiall caufe it to be refleded, or
luffer it to be tranfmitted. And, Thirdly, becaufe thofe
furfaces of tranfparent Bodies which have the greateft
refrading power, refled the greateft quantity of Light,
as was Shewed in the firft Propofition.

PROP. X.

If Light be fwifter in Bodies than in Vacuo in the
proportion of the Sines which measure the refraffion of the
Bodies , the forces of the Bodies to refleiB and refrafl Light ,

[ 7 1 3

are very nearly proportional to the denjities of the fame
Bodies , excepting that unduom and fulphureom Bodies re -
frad more than others of this fame denfity .

Let A B reprefent the refracting plane furface of any
Body, and IC a ray incident very obliquely upon the

Body in C, fo that the Angle A CI may be infinitely
little, and let CR be the refraCted ray. From a given
point B perpendicular to the refracting furface ereCt
B R meeting with the refrafted ray CR in R, and if
CR reprefent the motion of the refraCted ray, and this
motion be diftinguilhed into two motions CB and BR,
whereof CB is a parallel to the refraCting plane, and
BR perpendicular to it: CB {hall reprefent the motion
of the incident ray, and B R the motion generated by
the refraCtion, as Opticians have of late explained.

Now if any body or thing in moving through any
fpace of a giving breadth terminated on both tides by
two parallel plains, be urged forward in all parts ofi
that {pace by forces tending direCtly forwards towards
the laft plain, and before its incidence on the firft
plane, had no motion towards it, or but an infinitly
little one *. and if the forces in all parts of that fpace, #
between the planes be at equal diftances from the planes
equal to one another, but at feveral diftances be bigger
or lefs in any given proportion, the motion generated-
by the forces in the whole paflage of the body or thing-

through.

[70

through that fpace lha.ll be in a fubduplicate proportion
of the forces, as Mathematicians will eafily underftand.
And therefore if the fpace of activity of the refracting
luperficies of the Body be confidered as fuch a fpace,
the motion of the ray generated by the refraCting force
of the Body, during its paflage through that fpace
that is the motion BR muft be in a fubduplicate
proportion of that refraCting force : I fay therefore that
the lquare of the Line B R, and by confequence the
refrading force of the Body is very nearly as the den-
fity of the fame Body. For this will appear by the fol-
lowingTable, wherein the proportion of the Sines which
meafure the refraxions of feveral Bodies, the fquare
of BR fuppofing CB an unite, the denfities of the
Bodies eftimated by their fpecifick gravities, and their
refraCtive power in refpeCt of their denfities are fet
down in feveral Columns.

The

[73]

The refrafting Bodies.

The Proportion
of the Sines of
incidence and
refraction of
yellow Light,

The Square of
B R, to which
the refracting
force oftheBo -
dy is propor¬
tionate.

The denfity
and fpeci -
fc gravity
of the Bo-
dy.

The refra¬
ctive power
of the Body
in refpeCt
of its den-
fity.

A Pfeudo-Topazius, be¬
ing a natural,pellucid,
brittle, hairy Stone, of

23 t0

^699

4’ 2 7

3979

a yellow Colour

385°

4160

Air

2851 to

o’ooo 5 2

o’oOI2 5

Glafs of Antimony

17 to

2*568

1*213

5*28

2*252

'4864

A Selenitis

61 to

5386

Glafs vulgar

3 1 to

1*4025

1 ’44 5

2 7 5 §

5436

Cryltal of the Rock

25 to

2^65

2? 7 2

2?I 43

5450

Hand Cryftal

1*778

6536

Sal Gemmae

17 to

1388

1T267

6477

Alume

3 5 to

*’ 7*4

i ? 9

6 57 °

Borax

22 tO

1*1511

6716

Niter

3 2 to

t ’345

1*295

1*041

7°79

Dantzick Vitriol

3°3 to

200

I ’ 7 I 5

7 5 5 *

Oyl of Vitriol

10 to

x *7

6124V

Rain Water

529 to

396

°’ 7 8 45

I * 1 79

7 S 45

Gumm Arabic

31 to

i’375

8574

Spirit of Wine well recti¬
fied

100 to

° J 8y6<y

o’866

10121

Camphire

3 to

T25

©’996

91 3

° y 93 2
0*874

1*04

3’4 -

12551

Oyl Olive

22 to

1*1511

12607

Lintfeed Oyl

40 to

1T948

12819

Spirit of Turpentine

25 to

1T626

13222

Ambar

14 to

I 7 42

4^949

i 3 6 54

A Diamond

IOO to

M 55 6

The refraftion of the Air in this Table is determined
by that of the Atmofphere obferved by Aftronomers.
For if Light pals through many refracting fubftances or
mediums gradually denfer and denier, and terminated

L1 with

' ' 3

with parallel furfaces, the fumm of all the refradions
wifi be equal to the Angle refradion which it would
have fullered in palling immediately out of the firft
medium into the laft. And this holds true, though tire
number of the refracting fubftances be increafed to infi¬
nity, and the diftances from one another as much de¬
creased^ fo that the Light may be refraded in every
point of its paffage, and by continual refradions bent
into a curve Line. And therefore the whole refradion
of Light in paffing through the Atmofphere from the
higheft and rareft part thereof down to the loweft and
denfeft part, mult be equal to the refradion which it
would futfer in paffing at like obliquity out of a Va¬
cuum immediately into Air of equal denfity with that
in the loweft part of the Atmofphere.

Now, by this Table, the refradions of a Pfeudo-To-
paz, a Selenitis, Rock Cryftal, Ifiand Cryftal, Vulgar
Glafs (that is, Sand melted together ) and Glafs of
Antimony, which are terreftrial ftony alcalizate con¬
cretes,and Air which probably arifes from fuch fubftances
by fermentation,though thefe be fubftances very differing
from one another in denfity, yet they have their refra¬
ctive powers almoft in the fame proportion to one ano¬
ther as their denfities are, excepting that the refradion of
that ftrange fubftance Iiland-Cryftal is a little bigger
than the reft. And particularly Air, which is 3 400 times
rarer than thePfeudo-Topaz, and 4200 times rarer than
Glafs of Antimony, has notwithftanding its rarity the
fame refradive power in refped of its denfity which
thofe two very denfe fubftances have in refped of theirs,
excepting fo tar as thofe two differ from one another.

Again,

[ 75 ]

Again, the refraCtion of Camphire, OyLOlive, Lint-
feed Oyl, Spirit of Turpentine and Amber, which are
fat fulphureous unCtuous Bodies, and a Diamond, which
probably is an unctuous fubftance coagulated, have their
refractive powers in proportion to one another as their
denfities without any confiderable variation* But the
refraCtive power of thefe unCtuous fubftances is two
or three times greater in refpeCi of their denfities than
the refraCtive powers of the former fubftances in refpeCfc
of theirs.

Water has a refraCtive power in a middle degree be¬
tween thole two forts of fubftances, and probably is of
a middle nature* For out of it grow all vegetable and
animal fubftances, which confift as well of fulphureous
fat and inflamable parts, as of earthy lean and alcalh
zate ones*

Salts and Vitriols have refraCtive powers in a middle
degree between thofe of earthy fubftances and Water,
and accordingly are compofed of thofe two forts of fub¬
ftances* For by diftillation and rectification of their
Spirits a great part of them goes into Water, and a great
part remains behind in the form of a dry fixt earth ca¬
pable of vitrification*

Spirit of Wine has a refraCtive power in a middle
degree between thofe of Water and oyly fubftances, and
accordingly feems to be compofed of both, united by
fermentation ; the Water, by means of fome faline Spi¬
rits with which ’tis impregnated, diffolving the Oyl,
and volatizing it by the aCtion. For Spirit of Wine is
inflamable. by means of its oyly parts, and being diftiL
led often from Salt of Tartar, grows by every diftilla¬
tion more and more aqueous and fiegmatick. And

L1 2 Chymifts

Chymifts obferve, that Yegitables (as Lavender, Rue,
Marjoram, i 3 c.) diftilled per fe , before fermentation
yield Oyls without any burning Spirits, but after fer¬
mentation yield ardent Spirits without Oyls : Which
fhews, that their Oyl is by fermentation converted into
Spirit. They find alfo, that if Oyls be poured in final!
quantity upon fermentating Vegetables, they diftil over
after fermentation in the form of Spirits.

So then, by the foregoing Table, all Bodies feem to
have their refractive powers proportional to their
denfities, (or very nearly ;) excepting fo far as they
partake more or lefs of fulphurous oyly particles, and
thereby have their refractive power made greater or
lefs. Whence it feems rational to attribute the refra¬
ctive power of all Bodies chiefly, if not wholly, to the
fulphurous parts with which they abound. For it’s
probable that all Bodies abound more or lefs with Sul¬
phurs. And as Light congregated by a Burning-glafs
aCts mod upon fulphurous Bodies, to turn them in¬
to fire and flame ; fo, fince all aCtion is mutual, Sul¬
phurs ought to aCt mod upon Light. For that the
aCtion between Light and Bodies is mutual, may appear
from this Confideration, That the denfeft Bodies which
refraCt and refleCt Light mod ftrongly grow hotteft in
the Summer-Sun, by the aCtion of the refraCted or re¬
flected Light.

I have hitherto explained the power of Bodies to re¬
flect and refraCt, and fliewed, that thin tranfparent
plates, fibres and particles do, according to their leveral
thicknefles and denfities, refleCt feveral forts of rays,
and thereby appear of feveral Colours, and by conle-
quence that nothing more is requifite for producing all

[ 77 ]

the Colours of natural Bodies than the leveral fizes and
denfities of their tranfparent particles. But whence it
is that thefe plates, fibres and particles do, according
to their feveral thickneffes and denfities, refledf federal
forts of rays, I have not yet explained. To give fome
infight into this matter, and make way for underftan-
ding the next Part of this Book, I fhali conclude this
Part with a few more Propofitions. Thofe which pre¬
ceded refpedt the nature of Bodies, thefe the nature of
Light : For both mu ft be under ft ood before the reafon
of their actions upon one another can be known. And,
becaufe the laft Propofition depended upon the velo¬
city of Light, I will begin with a Propofition of that
kind.

y c

PROP. XL

Light is propagated from luminous Bodies in time , and
fpends about feven or eight minutes of an hour in puffing
from the Sun to the Earth .

This was obferved firft by Romer , and then by others^
by means of the Eclipfes of the Satellites of Jupiter..,
For thefe Eclipfes, when the Earth is between the Sun
and Jupiter , happen about feven or eight minutes fooner
than they ought to do by the Tables, and when the Earth
is beyond the Sun they happen about feven or eight mi¬
nutes later than they ought to do; the reafon being, that
the Light of the Satellites has farther to go in the latter
cafe than in the former by the Diameter of the Earth's
Orbit. Some inequalities of time may arife from the
eccentricities of the Orbs of the Satellites; but thofe
cannot anfw.er in all the Satellites, and at all times

[ 78 ]

to the potition and diftance of the Earth from the Sun,
The mean motions of Jupiter's Satellites is alfo fwifter
in his defcent from his Aphelium to his Perihelium,
than in his afcent in the other half of his Orb : But this
inequality has no refpeCt to the pofition of the Earth,
and in the three interior Satellites is infenfible, as I find
by computation from the Theory of their gravity.

P R O P, XII.

Every ray of Light in its pajfage through any refra¬
cting furface is put into a certain tranfient conflitution
or f ate , which in the progrefs of the ray returns at
equal intervals , and difpojes the ray at every return
to be eafily tranfmitted through the next refratting fur-

face , and between the returns to be eafdy reflected by
it.

This is manifeft by the 5th, 9th, x 2th and 15th Ob-
fervations. For by thofe Obfervations it appears, that
one and the fame fort of rays at equal Angles of inci¬
dence on any thin tranlparent plate, is alternately refle¬
cted and tranfmitted for many fucceffions accordingly,
as the thicknefs of the plate increafes in arithmetical
progreffion of the numbers o, 1,2, 3,4., 5, 6, 7, 8, tev.
lo that if the firft reflexion (that which makes the firft
or innermoft of the Rings of Colours there defcribed )
be made at the thicknefs 1 ,the rays fhall be tranfmitted at
the thickneffes o, 2, 4, 6, 8, 10, 12, 3 $c. and thereby
make the central Spot and Rings of Light, which ap¬
pear by tranfmiflion, and be reflected at the thicknefs
S 3 ) 5 5 7 t V 11 ^- and thereby make the Rings which

appear

l 79 j

appear by reflexion. And this alternate reflexion and
tranftnifiion, as I gather by the o. 4th Obfervation, con¬
tinues for above an hundred viciffitudes , and by the
the Obfervations in the next part of this Book, for many
thoufands, being propagated from one furface of a Glafs-
plate to the other, though the thicknefs of the plate
be a quarter of an Inch or above : So that this alter¬
nation feems to be propagated from every refracting
furface to all diftances without end or limitation,

.This alternate reflexion and refraction depends on
both the furfaces of every thin plate., becaufe it de¬
pends on their diftance* By the aith Obfervation, if
either furface of a thin plate of Mufcovy-Glafs be wet¬
ted, the Colours caufed by the alternate reflexion
and refraCtion grow faint, and therefore it depends on
them. both.

It is therefore performed at the fecond furface, for
if it were performed at the firft, before the rays ar¬
rive at the fecond, it would not depend on the fe»
condL

It is alfo influenced by fame aCtion or difpofition,
propagated from the firft to the fecond, becaufe other-
wife at the fecond it would not depend on the firft. And
this aCtion or difpofition, in its propagation, intermits
and returns by equal intervals, becaufe in all its pro-
grefs it inclines the ray at one diftance from the firft
furface to be reflected by the fecond, at another to be
tranfmitted by it, and that by equal intervals for innu¬
merable viciffitudes. And becaufe the ray is difpofed
to reflexion at the diftances 1, 3, 5, 7, 9, If 1 c. and to
tranfmiffion at the diftances o, a, 4, 6, 8, 10, tec, ( for
its tranfmiffion through the firft furface, is at the di¬
ftance

[ So ]

fence o, and it is tranfmitted through both toge¬
ther, if their diftance be infinitely little or much lefs
than i) the difpofition to be tranfmitted at the diftances
a, 4, 6, 8, 10, is to be accounted a return of the
fame difpofition which the ray firft had at the diftance o,
that is at its tranlmiffion through the firft refraCting fur-
face. All which is the thing I would prove.

What kind of aCtion or difpofition this is ? Whether
it confift in a circulating or a vibrating motion of the
ray, or of the medium, or fomething elfe ? I do not
here enquire. Thofe that are averfe from affenting to
any new difcoveries, but fuch as they can explain by an
Hypothefts, may for the prefent fuppofe, that as Stones
by falling upon Water put the Water into an undula¬
ting motion, and all Bodies by percuffion excite vibra¬
tions in the Air; fo the rays of Light, by impinging on
any refraCting or reflecting furface, excite vibrations in
the refraCting or reflecting medium or fubftance, and
by exciting them agitate the folid parts of the refraCting
or reflecting Body, and by agitating them caufe the Body
to grow warm or hot; that the vibrations thus excited
are propagated in the refraCting or reflecting medium
or fubftance, much after the manner that vibrations are
propagated in the Air for caufing found, and move
fafter than the rays fo as to overtake them; and that
when any ray is in that part of the vibration which con-
fpires with its motion, it eafily breaks through a re¬
fraCting furface, but when it is in the contrary part of
the vibration which impedes its motion, it is eafily
reflected ; and, by conlequence, that every ray is fuc-
ceflively difpofed to be eafily reflected, or eafily tranf-
inttted, by every vibration which overtakes it. But

whether

[8ij

whether this Hypothefis be true or falfe I do not here
confider. I content my felf with the bare difcovery,
that the rays of Light are by fome caufe or other alter¬
nately difpofed to be refle&ed or refrafted for many vh
ciffitudes.

DEFINITION

The returns of the diffofition of any ray to he reflected
I will call its Fits of eafy reflexion, and tbofe of
its difpojition to be tranfmitted its Fits of eafy tranf-
miffion, and the Jpace it gaffes between every re¬
turn and the next return , the Interval of its
Fits.

PROP, XIII.

The reafon why the fur faxes of all thick trrnfparent
Bodies refleH fart of the Light incident on them , and
ref rail the reft , is^ that fome rays at their incidence are
in Fits
mijjion.

This may be gathered from the 24th Obfervation,
where the Light reflected by thin plates of Air and Glafs,
which to the naked Eye appeared evenly white all over
the plate, did through a Prifm appear waved with many
fucceffions of Light and Darknels made by alternate fits
of eafy reflexion, and eafy tranfmiffion 5 the Prifm
fevering and diftinguifhing the waves of which the
white reflected Light was compofed, as was explained
above.

of eafy reflexion , and others in Fits of eafy tranfl

And

i>]

And hence Light is in fits of eafy reflexion and eafy
tranfmiflion, before its incidence on tranfparent Bodies.
And probably it is put into fuch fits at its firft emiflion
from luminous Bodies, and continues in them during
all its progrefs, For thefe fits are of a lading Nature,
as will appear by the next part of this Book*

In this Propofition I fuppofe the tranfparent Bodies
to be thick, becaufe if the thicknefs of the Body be
much lefs than the interval of the fits of eafy reflexion
and tranfmiflion of the rays, the Body lofethits reflecting
power. For if the rays, which at their entering into
the Body are put into fits of eafy tranfmiflion, arrive at
the furtheft lurface of the Body before they be out of
thofe fits they muft be tranfmitted. And this is the
reafon why Bubbles of Water lofe their reflecting power
when they grow very thin, and why all opake Bo¬
dies when reduced into very fmall parts become tranf¬
parent.

PROP. XIV.

Thofe fur faces of tranfparent Bodies , 'which if the ray
he in a fit of refraction do refraCt it mofl ftrongly y if the
fay be in a fit of reflexion do refleCt it mofl eafily.

For we fhewed above in Prop. 8. that the caufe of
reflexion is not the impinging of Light on the folid
impervious parts of Bodies, but fome other pow r er by
which thofe folid parts a£t on Light at a diftance. We
fhewed alfo in Prop. 9. that Bodies reflect and refra£t
Light by one and the fame power varioufly exercifed in
various circumftances, and in Prop. 1. that the mofl:
ftrongly refracting furfaces reflect the mofl Light: All

which

which compared together evince and ratify both this
and the laft Propolition,

PROP, XV,

In any one and the fame fort of rays emerging in any
jingle out of any refracting fur face into one and the fame
medium , the interval of the following fits of eafy reflexion
and tranfmijflon are either accurately or very nearly , as
the Re Cl angle of the fee ant of the Angle of refraSion , and
of the fecant of mother Angle y whofe fine is the firfl of
106 arithmetical mean proportionals y between the fines
of incidence and refraction counted from the fine of re~
fradion .

This is manifeft by the 7th Oblervation.

PROP, XVL

In feveral forts of rays emerging in equal Angles out
of any refraCting fur face into the fame medium ,, the inter*
vals of the following fits of eafy reflexion and eafy tranfl
mijfion are either accurately , or very nearly , as the Cube*
roots of the Squares of the lengths of a Chord, which found
the notes in an Eighty fol, la, fa, fol r la, mi, fa, fol, with
all their intermediate degrees answering to the Colours of
thofe raysy according to the Analogy deferibed in the fe*
venth Experiment of the fecond Booh

This is manifeft by the 13 th and 14th Obfervations*

Mm 2

PROP,

[8+]

PROP. XVII.

If rays of any one fort pafs perpendicularly into feveral
mediums , the intervals of the fits of eafy reflexion and
tranfmijfion m any one medium , is to thofe intervals in
any other as the fine of incidence to the fine of refraHion ,
when the rays pafs out of the fir ft of thofe two mediums
into the fecond.

This is inanifeft by the loth Obfervatiom

PROP. XVIII.

If the rays which paint the Colour in the confine of
yellow and orange pafs perpendicularly out of any medium
into ylir^ the intervals of their fits of eafy reflexion are
the zf~- a th part of an Inch . iAnd of the fame length are
the intervals of their fits of eafy tranfmijfion.

This is manifeft by the 6th Qbfervation.

From thefe Propofitions it is eafy to colled the in¬
tervals of the fits of eafy reflexion and eafy tranfmif-
fion of any fort of rays refraded in any Angle into
any medium, and thence to know, whether the rays
Ihall be refleded or tranfmitted at their fubfequent
incidence upon any other pellucid medium. Which
thing being ufeful for underflanding, the next part of
this Book was here to be fet down. And for the lame
-reafon I add the two following Propofitions.

PROP.

[85]

PROP. XIX.

If any fort of rays falling on the polite fur face of am
pellucid medium be reflecled back^ the fits of eafy re¬
flexion which they have at the point of reflexion , J,hall
fiill continue to return , and the returns Jhall be at di¬
fiances from the point of reflexion in the arithmetical
progreflion of the numbers 2, 4, 6, 8, 10, I2,&c. and be¬
tween thefe fits the rays Jhall be in fits of eafy tranfl
mijflon .

For fince the fits of eafy reflexion and eafy tranf-
miflion are of a returning nature, there is no reafbn
why thefe fits, which continued till the ray arrived at
the reflecting medium, and there inclined the ray to
reflexion, fliould there ceafe, And if the ray at the
point of reflexion was in a fit of eafy reflexion, the
progreflion of the diftances of thefe fits from that point
•rnuft begin from o, and fo be of the numbers o, a, 4,
6, 8, 1 Sc.. And therefore the progreflion of the di¬
ftances of the intermediate fits of eafy tranfmiflion rec¬
koned from the fame point, muft be in the progreflion
of the odd numbers 1, 4, 5, 7, 9, If 1 c, contrary to what
happens when the fits are propagated from points of
refraction,

PRO P. XX

The intervals of the fits of eafy reflexion and eajf
tranfmiflion , propagated from points of reflexion into any
medium , are equal to the intervals of the like fits which
the fame rays would have , if refracted into the fame

medium

[ 8 < 5 ]

medium in Angles of refraction equal to their Angles of
reflexion .

For when Light is reflected by the fecond furface of
thin plates, it goes out afterwards freely at the firft fur-
face to make the Rings of Colours which appear by
reflexion, and by the freedom of its egrefs, makes the
Colours of thefe Rings more vivid and ftrong than thofe
which appear on the other fide of the plates by the
tranfmitted Light. The reflefted rays are therefore in
fits of eafy tranfmiffion at their egrefs ; which would
not always happen, if the intervals of the fits within
the plate after reflexion were not equal both in length
and number to their intervals before it. And this confirms
alfo the proportions fet down in the former Propofition.
For if the rays both in going in and out at the firft furface
be in fits of eafy tranfmiffion, and the intervals and num¬
bers of thofe fits between the firft and fecond furface,
before and after reflexion, be equal • the diftances or
the fits of eafy tranfmiffion from either furface, muff be
in the fame progrefiion after reflexion as before; that
is, from the firft furface which tranfmitted them, in
the progrefiion of the even numbers o, 2, 4, 6, 8, ) 3 c.
and from the fecond which reflected them, in that of
the odd numbers 3^ 5^ 7^ &V* But thefe two Pro-
pofitions will become much more evident by the Gbfer-
various in the following part of this Book*

THE

SECOND BOOK

OPTICK S.

PART IV.

Observations concerning the Reflexions and Colours of
thick tranfparent folijhed 'plates.

T Here is no Glafs or Speculum how well foever
polifhed, but, befides the Light which it refracts
or refle&s regularly, fcatters every way irregularly a
faint Light, by means of which the polifhed furface,
when illuminated in a dark Room by a beam of the
Sun’s Light, may be eafily feen in all pofitions of the
Eye. There are certain Phenomena of this fcattered
Light, which when I firft obferved them, feemed very
ft range and furprifing to me. My Operations were
as follows.

OBS.

[ S8 ]

O B S. I.

The Sun firming into my darkened Chamber through
a Hole ] of an Inch wide, I let the intromitted beam
of Light fall perpendicularly upon a Glafs Speculum
ground concave on,one fide and convex on the other,
td a Sphere of five Feet and eleven Inches Radius, and
quickTilvered over on the convex fide. And holding
a white opake Chart, or a Quire of Paper at the Center
of the Spheres to which the Speculum was ground, that
is, at thediftance of about five Feet and eleven Inches
from the Speculum, in fuch manner, that the beam of
Light might pafs through a little Hole made in the
middle of the Chart to the Speculum, and thence be
reflected back to the fame Hole : I obferved upon the
Chart four or five concentric Irifes or Rings of Colours,
like Rain-bows, encompaffing the Hole much after the
manner that thofe, which in the fourth and following
Obfervations of the firftpart of this third Book appeared
between theObje<ft-GlafiTes,encompafTed the black Spot,
but yet larger and fainter than thofe. Thefe Rings as
they grew larger and larger became diluter and fainter,
■lb that the fifth was fcarce vifible. Yet fometimes,
when the Sun fhone very clear, there appeared faint
Lineaments of a fixth and feventh. If the diftance of
the Chart from the Speculum was much greater or much
lefs than that of fix Feet, the Rings became dilute and
vanifhed. And if the diftance of the Speculum from
the Window was much greater than that of fix Feet,
the reflected beam of Light wou ld be fo broad at the
diftance of fix Feet from the Speculum where the Rings

appeared,

appeared, as to obfcure one or two of the mnermoft
Rings, And therefore I ufually placed the Speculum
at about fix Feet from the Window; fo that its Focus
might there fall in with the center of its concavity at the
Rings upon the Chart, And this pofture is always to
be underftood in the following Oblervations where no
other is expreft,

O B S. IL

The Colours of thefe Rain-bows fucceeded one ano-
ther from the center outwards, in the fame form and
order with thofe which were made in the ninth Obfer-
vation of the firft Part of this Book by Light not re¬
flected, but tranfmitted through the twoObjeCLGlaffes*
For, firft, there was in their common center a white
round Spot of faint Light, fomething broader than the
reflected beam of Light; which beam fometimes fell
upon the middle of the Spot, and fometimes by a little
inclination of the Speculum receded from the middle,,
and left the Spot white to the center.

This white Spot was immediately encompaffed with
a dark grey or ruffet, and that darknefs with the Co¬
lours of the firft Iris, which were on the infide next
the darknefs a little violet and indico, and next to that
a blue, which on the outfide grew pale, and then fuc¬
ceeded a little greenifti yellow, and after that a brighter
yellow, and then on the outward edge of the Iris a red,
which on the outfide inclined to purple.

This Iris was immediately encompaffed with a fe¬
cund, whole Colours were in order from the infide

N n out-

[ 9 ° ]

outwards, purple, blue, green, yellow, light red, a red
mixed with purple.

Then immediately followed the Colours of the third
Iris, which were in order outwards a green inclining
to purple, a good green, and a red more bright than
that ot the former Iris.

The fourth and fifth Iris feemed of a bluiffi green
within, and red without, but fo faintly that it was dif¬
ficult to difcern the Colours.

OBS. III.

Meafuring the Diameters of thefe Rings upon the
Chart as accurately as I could, I found them alfo in
the fame proportion to one another with the Rings
made by Light tranfmitted through the two Objeft-
Glaffes. For the Diameters of the four firft of the
bright Rings meafured between the brighteft parts of
their orbits, at the diftance of fix Feet from the Specu¬
lum were ajy, 3! Inches, whofe fquares are in

arithmetical progreffion of the numbers 1, a, 4. If
the white circular Spot in the middle be reckoned
amongft the Rings, and its central Light, where it
feemstobe moft luminous, be put equipollent to an
infinitely little Ring; the fquares of the Diameters of the
Rings will be in the progreffion o, 1, a, 3, 4, 5 f 1 c. I
meafured alfo the Diameters of the dark Circles be¬
tween thefe luminous ones, and found their fquares
in the progreffion of the numbers i \ 9 3-, foV.

the Diameters of the firft four at the diftance of fix Feet
from the Speculum, being 2 ~ 6 , 3^ Inches. If

the diftance of the Chart from the Speculum was in-

creafed

[90

creafed or diminilhed, the Diameters of the Circles were
increafed or diminifhed proportionally.

OBS. IV.

By the analogy between thefe Rings and thofe de-
fcribed in the Obfervations of the firftPart of this Book,
I fufpeCted that there were many more of them which
fpread into one another, and by interfering mixed their
Colours, and diluted one another fo that they could
not be feen apart. I viewed them therefore through a
Prifm, as I did thofe in the 24th Obfervation of the
firft Part of this Book. And when the Prifm was fo
placed as by refracting the Light of their mixed Co¬
lours to feparate them, and diftinguilh the Rings from
one another, as it did thofe in that Obfervation, I could
then fee them diftinCter than before, and eafily num¬
ber eight or nine of them, and fometimes twelve or
thirteen. And had not their Light been fo very faint,
I queftion not but that I might have feen many more.

OBS. V.

Placing a Prifm at the Window to refraCt the intro¬
mitted beam of Light, and caft the oblong Spectrum
of Colours on the Speculum : I covered the Speculum
with a black Pap£r which had in the middle of it a Hole
to let any one of the Colours pafs through to the Spe¬
culum, whilft the reft were intercepted by the Paper.
And now I found Rings of that Colour only which fell
upon the Speculum. If the Speculum was illuminated
with red the Rings were totally red with dark inter-

N n 2 vats,

[ 92 ]

vals, if with blue they were totally blue, and fo of the
other Colours. And when they were illuminated with
any one Colour, the Squares of their Diameters mea-
fured between their moft luminous parts, were in the
arithmetical progreffion of the numbers o, i, a, 3,4, and
the Squares of the Diameters of their dark intervals in
the progreffion of the intermediate numbers ii, 3I;
But if the Colour was varied they varied their magni¬
tude. In the red they were largeft, in the indico and
violet leaft, and in the intermediate Colours yellow,
green and blue ; they were of leveral intermediate big-
neffes anlwering to the Colour, that is, greater in yel¬
low than in green, and greater in green than in blue.
And hence I knew that when the Speculum was illumi¬
nated with white Light, the red and yellow on the out-
fide of the Rings were produced by the leaft refrangible
rays, and the blue and violet by the moft refrangible,
and that the Colours of each Ring fjpread into the Co¬
lours of the neighbouring Rings on either fide, after
the manner explained in the firft and fecond Part of this
Book, and by mixing diluted one another fo that they
could not be diftinguiftied, unlefs near the center where
they were leaft mixed. For in this Obfervation I could
fee the Rings more diftinftly, and to a greater number
than before, being able in the yellow Light to number
eight or nine of them, befides a faint ffiadow of a tenth.
To fatisfy my felf how much the Colours of the feveral
Rings fpread into one another, I meafured the Diame¬
ters of the fecond and third Rings, and found them
when made by the confine of the red and orange to be
the fame Diameters when made by the confine of blue
and indico, as 9 to 8, or thereabouts. For it was hard

to,

[P3 ]

to determine this proportion accurately. Alfo the Cir¬
cles made fucceffively by the red., yellow and green,
differed more from one another than thofe made fuccef-
lively by the green., blue and indico. For the Circle
made by the violet was too dark to be feen. To carry
on the computation, Let us therefore fuppofe that the
differences of the Diameters of the Circles made by the
outmoft red, the confine of red and orange, the confine
of orange and yellow, the confine of yellow and green,
the confine of green and blue, the confine of blue and ?
indico, the confine of indico and violet, and outmoft vio¬
let, are in proportion as the differences of the lengths
of a Monochord which found the tones in an Eight ;
fol, la^ fa^ fol, la^ mi^ fa y [ol y that is, as the numbers t
Jg, L k, l 7 , [j, L And if the Diameter of the Circle made
by the confine of red and orange be 9 A, and that of
the Circle made by the confine of blue and indico be
8 A as above, their difference 9 A —— 8 A will be to
the difference of the Diameters of the Circles made by
the outmoft red, and by the confine of red and orange,
as i 8 + + K + if to t, that is as h to i or 8 to 3, and to

the difference of the Circles made by the outmoft violet,
and by the confine of blue and indico, as h +f» +.7* -f § 7
to h +L, that is, as L to f 4 , or as 16 to 5, And there¬
fore thefe differences will be I A and U A. Add the
fir ft to 9 A and fubduft the laft from 8 A, and yow
will have the Diameters of the Circles made by the
leaft and moft refrangible rays V A and pi A; Thefe
Diameters are therefore to one another as 75 to 6i - on
50 to 41, and their Squares as acoo to 1681, that Is,
as 3 to 2 very nearly. Which proportion differs not
much from the proportion of the Diameters of the

Circles-*

[9+]

Circles made by the outmoft red and outmoft violet in
the 13 th Obfervation of the firft part of this Book.

O B S. VI.

Placing my Eye where thefe Rings appeared plaineft,
I faw the Speculum tinged all over with waves of Co¬
lours ( red, yellow, green, blue ;) like thole which in
the Obfervations of the hr ft Part of this Book appeared
between the Objed-GlalTes and upon Bubbles of Water,
but much larger. And after the manner of thofe, they
were of various magnitudes in various pohtions of the
Eye, fweiling and fhrinking as I moved my Eye this
way and that way. They were formed like Arcs of
concentrick Circles as thofe were, and when my Eye
was over againft the center of the concavity of the Spe¬
culum (that is, 5 Feet and 10 Inches diftance from the
Speculum) their common center was in a right Line
with that center of concavity, and with the Hole in the
Window. But in other poftures of my Eye their center
had other pohtions. They appeared by the Light of
the Clouds propagated to the Speculum through the
Hole in the Window, and when the Sun Ihone through
that Hole upon the Speculum, his Light upon it was
of the Colour of the Ring whereon it fell, but by its
fplendor obfcured the Rings made by the Light of the
Clouds, unlefs when the Speculum was removed to a
great diftance from the Window, fo that his Light upon
it might be broad and faint. By varying the polition of
my Eye, and moving it nearer to or farther from the
diredt beam of the Sun’s Light, the Colour of the Sun’s
ceflefited Light conftantly varied upon the Speculum,

as it did upon my Eye, the fame Colour always ap¬
pearing to a By-ftander upon my Eye which to me ap¬
peared upon the Speculum. And thence I knew that
the Rings of Colours upon the Chart were made by thefe
reflected Colours propagated thither from the Specu¬
lum in feveral Angles, and that their production de¬
pended not upon the termination of Light and Shad-
dow*

O B S. VII,

By the Analogy of all thefe Phenomena with thofe of
the like Rings of Colours defcribed in the firft Part of
this Book, it teemed to me that thefe Colours were
produced by this thick plate of Glafs, much after the
manner that thofe were produced by very thin
plates. For, upon tryal, I found that if the Quick-
Ill ver were rubbed off from the back-fide of the Specu¬
lum, the Glafs alone would caufe the fame Rings, of
Colours, but much more faint than before ; and there¬
fore the Phenomenon depends not upon the Quick-
filver, unlefs fo far as the Quick-fiver by the increafing
the reflexion of the back-fide of the Glafs increafes the
Light of the Rings of Colours. I found alfo that a Spe¬
culum of metal without Glafs made fome years fince
for optical ufes, and very well wrought, produced none
of thofe Rings; and thence I underftood that thefe
Rings arife not from one fpecular furface alone, but
depend upon the two furfaces of the plate of Glafs where¬
of the Speculum was made, and upon the thicknefs of
the Glafs between them. For as in the yth and 19th,
Obfervations of the firft Part of this Book a thin plate

~ ~ ~ of

[5>«]

of Air, Water, or Glafs of an even thicknefs appeared
of one Colour when the rays were perpendicular to it,
of another when they were a little oblique, of another
when more oblique, of another when ftill more oblique,
and fo on ; fo here, in the lixth Obfervation, the Light
which emerged out of the Glafs in feveral obliquities,
made the Glafs appear of feveral Colours, and being
propagated in thofe obliquities to the Chart, there pain¬
ted Rings of thofe Colours. And as the reafon why a
thin plate appeared of feveral Colours in feveral obli¬
quities of the rays,was,that the rays of one and the fame
fort are reflected by the thin plate at one obliquity and
tranfmitted at another, and thofe of other forts tranf-
mitted where thefe are reheated, and reflected where
thefe are tranfmitted : So the reafon why the thick
plate of Glafs whereof the Speculum was made did ap¬
pear of various Colours in various obliquities, and in
thofe obliquities propagated thofe Colours to the Chart,
was, that the rays of one and the fame fort did at one
obliquity emerge out of the Glafs, at another did not
emerge but were reflected back towards the Quick-fil-
ver by the hither furface of the Glafs, and accordingly
as the obliquity became greater and greater emerged
and were relieved alternately for many fucceffions, and
that in one and the fame obliquity the rays of one fort
were reflected, and thofe of another tranfmitted. This
is manifeft by the firft Obfervation of this Rook : For
in that Obfervation, when the Speculum was illumi¬
nated by any one of the prifmatick Colours, that Light
made many Rings of the fame Colour upon the Chart
with dark intervals, and therefore at its emergence out
of the Speculum was alternately tranfmitted, and not

tranf-

[ 9 ? 1

tranfmitted from the Speculum to the Chart for many
fucceffions, according to the various obliquities of its
emergence. And when the Colour caft on the Specu¬
lum by the Prifm was varied, the Rings became of
the Colour caft on it, and varied their bignefs with their
Colour, and therefore the Light was now alternately
tranfmitted and not tranfmitted from the Speculum to
the Lens at other obliquities than before. It feemed to
me therefore that thefe Rings were of one and the fame
original with thofe of thin plates, but yet with this
difference that thofe of thin plates are made by the al¬
ternate reflexions and tranfmiffions of the rays at the
fecond furface of the plate after one paffage through it:
But here the rays go twice through the plate before
they are alternately reflected and tranfmitted ; fir ft,
they go through it from the firft furface to the Quick-
filver, and then return through it from the Quick-filver
to the firft furface, and there are either tranfmitted to
the Chart or reflected back to the Quick-filver, ac¬
cordingly as they are in their fits of eafie reflexion or
tranfmiffion when they arrive at that furface. For the
intervals of the fits of the rays which fall perpendicu¬
larly on the Speculum, and are reflected back in the
fame perpendicular Lines, by reafon of the equality of
thefe Angles and Lines,are of the fame length and num-
ber within the Glafs after reflexion as before by the
19th Propofition of the third Part of this Book. * And
therefore ftnce all the rays that enter through the firft
furface are in their fits of eafy tranfmiffion at their en¬
trance, and as many of thefe as are reflected by the fe¬
cond are in their fits of eafy reflexion there, all thefe
mull be again in their fits of eafy tranfmiffion at their

Q o return

return to the firft, and by confequence there go out of
the Glafs to the Chart., and form upon it the white
Spot of Light in the center of the Rings. For the rea-
fon holds good in all forts of rays , and therefore all
forts mu ft go out promifcuoully to that Spot, and by
their mixture caufe it to be white. But the intervals
of the fits of thole rays which are reflected more ob¬
liquely than they enter, muft be greater after reflexion
than before by the 15 th and 20th Prop. And thence
it may happen that the rays at their return to the firft
furface, may in certain obliquities be in fits of eafy re¬
flexion, and return back to the Guick-filver, and in
other intermediate obliquities be again in fits of eafy
tranfmiflion, and fo go out to the Chart, and paint on
it the Rings of Colours about the white Spot. And
becaufe the intervals of the fits at equal obliquities are
greater and fewer in the lefs refrangible rays, and lefs
and more numerous in the more refrangible, therefore
the lefs refrangible at equal obliquities fhall make fewer
Rings than the more refrangible, and the Rings made
by thofe fhall be larger than the like number of Rings
made by thefe ; that is, the red Rings fhall be larger
than the yellow, the yellow than the green, the green
than the blue, and the blue than the violet, as they
were really found to be in the 5th Obfervation. And
therefore the firft Ring of all Colours incompaffing the
white Spot of Light fhall be red without and violet
within, and yellow, and green, and blue in the middle,
as it was found in the fecond Obfervation; and thefe
Colours in the fecond Ring, and thofe that follow fhall
be more expanded till they fpread into one another,
and blend one another by interfering*

Thefe

[99 3 i

Thefe feem to be the reafons of thefe Rings in ge¬
neral, and this put me upon obferving the thicknefs of
the Glafs, and confidering whether the dimenfions and
proportions of the Rings may be truly derived from it
by computation.

O B S. VIII.

I meafured therefore the thicknefs of this concavo-
convex plate of Glafs, and found it every-where i of an
Inch precifely. Now, by the 6th Obfervation of the
firft Part of this Book, a thin plate of Air tranfmits the
brighteft Light of the firft Ring, that is the bright yel¬
low, when its thicknefs is the 8 ^- 0 th part of an Inch,
and by the i oth Obfervation of the fame part, a thin
plate of Glafs tranfmits the fame Light of the fame Ring
when its thicknefs is lefs in proportion of the fine of
refraction to the fine of incidence, that is, when its
thicknefs is the ~ath or ,- 3 ;—th part of an Inch, fup-
pofing the fines are as 11 to 17. And if this thicknefs
be doubled it tranfmits the fame bright Light of the
fecond Ring, if tripled it tranfmits that of the third,
and fo on, the bright yellow Light in all thefe cafes be¬
ing in its fits of tranfmiffion. And therefore if its thick¬
nefs be multiplied 34.386 times fo as to become^ of an
Inch it tranfmits the fame bright Light of the 34.386th
Ring. Suppofe this be the bright yellow Light tranf-
mitted perpendicularly from the reflecting convex fide
of the Glais through the concave fide to the white Spot
in the center of the Rings of Colours on the Chart: And
by a rule in the feventh Obfervation in the firft Part of
the firft Book, and by the 15 th and 20th Propofitions

O o 2 of

[ioo]

of the third Part of this Book, if the rays he made ob¬
lique to the Glafs, the thicknefs of the Glafs requh
fite to tranfmit the fame bright Light of the fame Ring
in any obliquity is to this thicknefs of of an Inch, ,as
the fecant of an Angle whole line is the firft of an hun¬
dred and fix arithmetical means between the fines of
incidence and refraction, counted from the fine of inci¬
dence when the refraction is made out of any plated Bo¬
dy into any medium incompafliog it, that is, in this cafe,
out of Glals into Air. Now if the thicknefs of the Glafs
be increafed by degrees,fo as to bear to its firft thicknefs,
( viz . that of a quarter of an Inch ) the proportions
which 343 86 (the number of fits of the perpendicular
rays in going through the Glafs towards the white Spot
in the center of the Rings,) hath to 34385, 34384,
34383 and 3438a (the numbers of thefits of the oblique
rays in going through the Glafs towards the firft, fe-
cond, third and fourth Rings of Colours,) and if the
firft thicknefs be divided into 100000000 equal parts,
the increafed thickneffes will be 100002908, 1000058165
100008725 and 10001163 3, and the Angles of which thefe
thickneffes are fecants will be 26' 13", 357' 5', 45' 6' and
5,a' a 6", the Radius being ioooooooo ; and the fines of
thefe Angles are 76a, 1079, 1321 and 1-5.25, and the
proportional fines of refraction 117a, 1659, 2031 and
2345, the Radius being 100000. For fince the fines
of incidence out of Glafs into Air are to the fines
of refraCtion as ri to 17, and to theabove-mentioned
fecants as. 11 to the firft of 106 arithmetical means
between 11 and 17,, that is as 11 to 11^, thefe fe¬
cants will he to the fines of refraCtion as 11-5 to 17,
and by this Analogy will give thefe fines. So then

.[ 101 ]

if the obliquities of the rays to the concave furface of
the Glals be fuch that the lines of their relradtion in
palling out of the Glafs through that furface into the
Air be 117:2, 1659, 2031, 2345, the bright Light of
the 34386th Ring lhall emerge at the thicknefles of the
Glafs which are to I of an Inch as 34386 to 34385,
34384, 34383, 34382, refpedtively. And therefore if
the thicknels in all thefe cafes be - of an Inch (as it is in
the Glafs of which the Speculum was made) the bright
Light of the 34385th Ring lhall emerge where the line
of refraction is 1172,and that of the 34384th,384383th
and 34382th Ring where the line is 1659, 2031, and
2345 refpedtively. And in thefe Angles of refraction
the Light of thefe Rings lhall be propagated from the
Speculum to the Chart, and there paint Rings about the
white central round Spot of Light which we laid was
the Light of the 34386th Ring. And the Semidiame¬
ters of thefe Rings lhall fubtend the Angles of refraCtion
made at the concave, lurface of the Speculum, and by
confequence their Diameters lhall be to the diftance of
the Chart from the Speculum as thole lines ofrefratition
doubled are to the Radius that is as 1172, 1659, 203 r,
and 2.345, doubled are to 100000. And therefore if,
the diftance of the Chart from the concave furface of
the Speculum be fix Feet (as it was in the third of thefe
Obfervations) the Diameters of the Rings of this bright
yellow Light upon the Chart lhall be x 7 688?

2 ’ 9 2 5 j F 375 Inches : For thefe Diameters are to A Feet
as the above-mentioned fines doubled are to the Radius.
No thefe Diameters of the bright yellow Rings, thus
found by computation are the very fame with thole
found in the third of thefe Obfervations by meafuring

therrq

[ 102 ]

them, (viz. with ijj. a|> a|i, and 3’-Inches, and there¬
fore the Theory of deriving thefe Rings from the thick-
nefs of the plate of Glafs of which the Speculum was
made, and from the obliquity of the emerging rays agrees
with the Obfervation. In this computation I have
equalled the Diameters of the bright Rings made by
Light of all Colours, to the Diameters of the Rings
made by the bright yellow. For this yellow makes the
brighteft part of the Rings of all Colours. If you defire
the Diameters of the Rings made by the Light of any
other unmixed Colour, you may find them readily by
putting them to the Diameters of the bright yellow ones
in a lubduplicate proportion of the intervals of the fits
of the rays of thole Colours when equally inclined to
the refracting or reflecting furface which caufed thofe
fits, that is, by putting the Diameters of the Rings made
by the rays in the extremities and limits of the feven
Colours, red, orange, yellow, green, blue, indico, violet,
proportional the Cube-roots of the numbers, 1, f, |, *,
>, *, %, {, which exprefs the lengths of a Monochard
founding the notes in an Eight: For by this means the
Diameter of the Rings of thefe Colours will be found
pretty nearly in the fame proportion to one another,
which they ought to have by the fifth of thefe Obfer-
vations.

And thus I fatisfied my felf that thefe Rings were of
the fame kind and original with thofe of thin plates,
and by conlequence that the fits or alternate difpofi-
tions of the rays to be reflected and tranfmitted are pro¬
pagated to great diftances from every reflecting and re¬
fracting furface. But yet to put the matter out of doubt
I added the following Obfervation.

OBS.

C io 3 ]

O B S. IX.

If thefe Rings thus depend on the thicknefs of the plate
ofGlafs their Diameters at equal di fiances from feveral
Speculums made of luch concavo-convex plates of Glals
as are ground on the fame Sphere, ought to be recipro¬
cally in a fubduplicate proportion of the thickneffes of
the plates of Glals. And if this proportion be found
true by experience it will amount to a demonftration-
that thefe Rings (like thofe formed in thin plates) do-
depend on the thicknefs of the Glafs. I procured there¬
fore another concavo-convex plate of Glafs ground on
both fides to the fame Sphere with the former plate :
Its thicknefs was f 2 parts of an Inch; and the Diameters
of the three firft bright Rings meafured between the
brighteft parts of their orbits at the diftance of 6 Feet
from the Glafs were 3. 4.5. 5k Inches. Now the thick¬
nefs of the other Glafs being of an Inch was to thick¬
nefs of this Glafs a s j to f, > that is as 31 to 10, or
3 rooooooo to iooqoooo © 5 and the roots of thefe numbers
are 17607 and 10000, & in the proportion of the firft
of thefe roots to the fecond are the Diameters of the
bright Rings made in this Obfervation by the thinner
Glafs, 3. 4k 5; to the Diameters of the fame Rings made
in the third of thel'e Oblervations by the thicker Glafs
i[k af that is, the Diameters of the Rings are reci¬
procally in a fubduplicate proportion of thickneffes of
the plates of Glafs.

So then in plates of Glafs which are alike concave on
one fide, and alike convex on the other fide, and alike
quick-filvered on the convex fides, and differ in nothing

but

[I°+]

but their thicknefs, the Diameters of the Rings are re¬
ciprocally in a fubduplicate proportion of the thickneffes
of the plates. And this fhews fufficiently that the Rings
depend on both the furfaces of the Glafs. They de¬
pend on the convex furface becaufe they are more lu¬
minous when that furface is quick-filvered over than
when it is without Quick-lilver. They depend alfo
upon the concave furface, becaufe without that furface
a Speculum makes them not. They depend on both
furfaces and on the diftances between them., becaufe
their bignefs is varied by varying only that diftance.
And this dependance is of the fame kind with that
which the Colours of thin plates have on the diftance
of the furfaces of thofe plates, becaufe the bignefs
of the Rings and their proportion to one another
and the variation of their bignefs arifing from the varia¬
tion of the thicknefs of the Glafs, and the orders of
their Colours, is fuch.as ought to refult from the Propo-
fitions in the end of the third Part of this Book, derived
from the the Phenomena of the Colours of thin plates
fet down in the firft Part.

There are yet other Phenomena of thefe Rings of
Colours but fuch as follow from the fame Propofitions
and therefore confirm both the truth of thofe Propofi¬
tions, and the Analogy between thefe Rings and the
Rings of Colours made by very thin plates. I fhall
fubjoyn fome of them.

t 10 $ ]

O B S. X.

When the beam of the Sun’s Light was reflected back
from the Speculum not direftly to the Hole in the Win¬
dow, but to a place a little diftant from it, the common
center of that Spot, and of all the Rings of Colours fell
in the middle way between the beam of the incident
Light, and the beam of the reflected Light, and by
confequence in the center of the fpherical concavity of
the Speculum, whenever the Chart on which the Rings
of Colours fell was placed at that center. And as the
beam of reflected Light by inclining the Speculum re¬
ceded more and more from the beam of incident Light
and from the common center of the coloured Rings be¬
tween them, thofe Rings grew bigger and bigger, and
fo alfo did the white round Spot,and new Rings of Co¬
lours emerged fucceffively out of their common center,
and the white Spot became a white Ring encompaffing
them ; and the incident and reflected beams of Light
always fell upon the oppoiite parts of this Ring, illumi¬
nating its perimeter like two mock Suns in the oppoiite
parts of an Iris. So then the Diameter of this Ring,
meafured from the middle of its Light on one fide to
the middle of its Light on the other fide, was always
equal to the diftance between the middle of the Incident
beam of Light, and the middle of the reflected beam
meaiured at the Chart on which the Rings appeared:
And the rays which formed this Ring were reflected by
the Speculum in Angles equal to their Angles of inci¬
dence, and by confequence to their Angles of refrafif ion
at their entrance into the Glafs, but yet their Angles of

' P p reflexion

[io6]

reflexion were not in the fame planes with their Angles
of incidence.

O B S. XL

The Colours of the new Rings were in a contrary
order to thofe of the former, and arofe after this man¬
ner. The white round Spot of Light in the middle of
the Rings continued white to the center till the diftance
of the incident ond reflected beams at the chart was
about | parts of an Inch, and then it began to grow
dark in the middle. And when that diftance was about
if 6 of an Inch, the white Spot was become a Ring en-
compafling a dark round Spot which in the middle in¬
clined to violet and indico. And the luminous Rings
incompaffing it were grown equal to thofe dark ones
which in the four firft Obfervations encompaffed them,
that is to fay, the white Spot was grown a white Ring
equal to the firft of thofe dark Rings, and the firft of
thofe luminous Rings was now grown equal to the fe-
cond of thofe dark ones, and the fecond of thofe lumi¬
nous ones to the third of thofe dark ones, and fo on.
For the Diameters of the luminous Rings were now

Inches.

When the diftance between the incident and reflected
beams of Light became a little bigger, there emerged
out of the middle of the dark Spot after the indico a
blue, and then out of that blue a pale green, and foon
after a yellow and red. And when the Colour at the
center was brighteft, being between yellow and red,
the bright Rings were grown equal to thofe Rings which
in the four firft Obfervations next encompaffed them;

that

[I° 7 3

that is to fay, the white Spot in the middle of thofe
Rings was now become a white Ring equal to the firft
of thofe bright Rings, and the firft of thofe bright ones
was now become equal to the fecond of thofe, and fo
on. For the Diameters of the white Rings, and of the
other luminous Rings incompaffing it, were now lit
ai ? aU, ^8,2or thereabouts.

When the diftance of the two beams of Light at the
Chart was a little more increafed, there emerged out
of the middle in order after the red, a purple, a blue,
a green, a yellow, and a red inclining much to purple,
and when the Colour was brighteft being between yel¬
low and red, the former indico, blue, green, yellow and
red, were become an Iris or Ring of Colours equal
to the firft of thofe luminous Rings which appeared in
the four firft Qbfervations, and the white Ring which
was now become the fecond of the luminous Rings was
grown equal to the fecond of thofe, and the firft of
thofe which was now become the third Ring was be¬
come the third of thofe, and fo on. For their Diame¬
ters were i*«, at, ? gi Inches, the diftance of the

two beams of Light, and the Diameter of the white
Ring being a- Inches.

When thefe two beams became more diftant there
emerged out of the middle of the purplifti red, firft a
darker round Spot, and then out of the middle of that
Spot a brighter. And now the former Colours (purple,
blue, green, yellow, and purplifti red) were become a
Ring equal to the firft of the bright Rings mentioned in
the four firft Qbfervations, and the Ring about this
Ring were grown equal to the Rings about that re*-
fpe&ively ; the diftance between the two beams of

P p ^ Light

[i°8]

Light and the Diameter of the white Ring (which
was now become the third Ring) being about 3. In¬
ches.

The Colours of the Rings in the middle began now
to grow very dilute, and if the diftance between the
two beams was increafed half an Inch, or an Inch more,
they vanifhed whilft the white Ring, with one or two
of the Rings next it on either fide, continued ftili vi-
fible. But if the diftance of the two beams of Light
was ftili more increafed thefe alfo vanifhed : For the
Light which coming from feveral parts of the Hole in
the Window fell upon the Speculum in feveral Angles of
incidence made Rings of feveral bigneffes, which diluted
and blotted out one another, as I knew by intercepting
fome part of that Light. For if I intercepted that part
which was neareft to the Axis of the Speculum the
Rings would be lefs, if the other part which was re-
moteft from it they would be bigger.

OBS. XIL

When the Colours of the Prifm were caft fucceflively
on the Speculum, that Ring which in the two laft Ob>
fervations was white, was of the fame bignefs in all the
Colours, but the Rings without it were greater in the
green than in the blue, and ftili greater in the yellow,
and greateft in the red. And, on the contrary, the
Rings within that white Circle were lefs in the green
than in the blue, and ftili lefs in the yellow, and lea ft
in the red. For the Angles of reflexion of thofe rays
which made this Ring being equal to their Angles of
incidence, the fits of every reflected ray within, the Glafs

after

[lop]

after reflexion are equal in length and number to the
fits of the fame ray within the Glafs before its incidence
on the reflecting furface; and therefore fince all the rays
of all forts at their entrance into the Glafs were in a fit
of tranfmiffion, they were alfo in a fit of tranfmiflion at
their returning to the fame furface after reflexion ; and
by confequence were tranfmitted and went out to the
white Ring on the Chart. This is the reafon why that
Ring was of the fame bignefs in all the Colours, and
why in a mixture of all it appears white. But in rays
which are reflected in other Angles, the intervals pf the
fits of the leaft refrangible being greateft, make the
Rings of their Colour in their progress from this white
Ring, either outwards or inwards, increafe or decrease
by the greateft fteps • fo that the Rings of this Colour
without are greateft, and within leaft. And this is the
reafon why in the laft Obfervation, when the Specu¬
lum was illuminated with white Light, the exterior
Rings made by all Colours appeared red without and
blue within, and the interior blue without and red
within.

Thefe are the Phenomena of thick convexo-concave
plates of Glafs, which are every where of the fame
thicknefs. 1 here are yet other Phenomena when thefe
plates are a little thicker on one fide than on the
other, and others when the plates are more or lefs con-
cave than convex, or plano-convex, or double-convex.
For in all thefe cafes the plates make Rings of Colours,
but after various manners ; all which, fo far as I have
yet obferved, follow from the Propofitions in the end
of the third part of this Book, and fo confpire to con¬
firm the truth of thofe Propofitions. But the Pheno¬
mena,,

[no]

mena are too various, and the Calculations whereby
they follow from thofe Propofitions too intricate to be
here profecuted. I content my felf with having profe-
cuted this kind of Phenomena fo far as to difcover their
caufe, and by difcovering it to ratify the Propofitions
in the third Part of this Book,

OBS. XIIL

As Light reflected by a Lens quick-filvered on the
back-fide makes the Rings of Colours above de-
Icribed, fo it ought to make the like Rings of Colours
in palling through a drop of Water; At the firft re¬
flexion of the rays within the drop, fome Colours ought
to be tranfmitted, as in the cafe of a Lens, and others
to be reflected back to the Eye. For inftance, if the
Diameter of a fmall drop or globule of Water be about
the 500th part of an Inch, fo that a red-making ray in
palling through the middle of this globule has 150 fits
of eafy tranfmiflion within the globule, and that all the
red-making rays which are at a certain diftance from
this middle ray round about it have 249 fits within the
globule, and all the like rays at a certain further di-
fiance round about it have 24.8 fits, and all thofe at a
certain further diftance 247 fits, and fo on ; thefe con-
centrick Circles of rays after their tranfmiflion, falling
on a white Paper, will make concentrick rings of red
upon the Paper , fuppofing the Light which pafles
through one Angle globule ftrong enough to be fenfible.
And, in like manner, the rays of other Colours will
make Rings of other Colours. Suppofe now that in a
fair day the Sun Ihines through a thin Cloud of fuch

globules

[Ill]

globules of Water or Hail, and that the globules are all
of the fame bignefs,and the Sun feen through this Cloud
lhall appear incompaffed with the like concent rick Rings
of Colours, and the Diameter of the firft Ring of red
fhall be 7-* degrees, that of the fecond 1 o'- degrees, that
of the third 12 degrees 33 minutes. And accordingly
as the globules of Water are bigger or Ids, the Rings
fhall be lefs or bigger. This is the Theory, and expe¬
rience anfwers it.. For in June 169a. I law by reflexion
in a Veflel of ftagnating Water three Halos Crowns or
Rings of Colours about the Sun, like three little Rain¬
bows, concentrick to his Body. The Colours of the
firft or innermoft Crown were blue next the Sun, red
without, and white in the middle between the blue
and red. Thofe of the fecond Crown were purple and
blue within, and pale red without, and green in the
middle. And thofe of the third were pale blue with¬
in, and pale red without; thefe Crowns inclofed one
another immediately, fo that their Colours proceeded
in this continual order from the Sun outward: blue,
white, red ; purple, blue, green, pale yellow and red ;
pale blue, pale red. The Diameter of the fecond Crown
meafured from the middle of the yellow and red on one
fide of the Sun, to the middle of the fame Colour on
the other fide was degrees, or thereabouts. The Dia¬
meters of the firf^and third I had not time to meafure,
but that of the firft feemed to be about five or fix de¬
grees, and that of the third about twelve. The like
Crowns appear fometimes about the Moon; for in the
beginning of the year 1664, Febr. 19th at night, I faw
two fuch Crowns about her. The Diameter of the firft
or innermoft was about three degrees, and that of the

fecond.

[H2]

feeond about five degrees and an half. Next about the
Moon was a Circle of white, and next about that the
inner Crown which was of a bluifli green within next the
white, and of a yellow and red without, and next about
thefe Colours were blue and green on the infide of the
outward Crown, and red on the outfide of it. At the
fame time there appeared a Halo about 22 degrees 35*
diftant from the center of the Moon. It was Elliptical,
and its long Diameter was perpendicular to the Horizon
verging below fartheft from the Moon. I am told that
the Moon has fometimes three or more concentrick
Crowns of Colours incompafling one another next about
her Body. The more equal the globules of Water or
Ice are to one another, the more Crowns of Colours
will appear, and the Colours will be the more lively.
The Halo at the diftance of 22- degrees from the Moon
Ys of another fort. By its being oval and remoter from
the Moon below than above, I conclude, that it was
made by refraCtion in fome fort of Hail or Snow floating
in the Air in an horizontal Pofture, the refracting Angle
being about 58 or 60 degrees,

T H E

[ri3]

THE

THIRD BOOK

O F

O P T I C K s.

Obfervations concerning the Inflexions of the rays of Light ,

and the Colours made thereby .

G Rimaldo has informed us, that if a beam of the
Sun’s Light be let into a dark Room through a
very fmall Hole, the fliadows of things in this Light
will be larger than they ought to be if the rays went
on by the Bodies in Freight Lines, and that thefe Ilia-
dows have three parallel fringes, bands or ranks of co¬
loured Light adjacent to them. But if the Hole be
enlarged the fringes grow broad and run into one ano¬
ther, lo that they cannot be diftinguifhed. Thefe broad
fhadows and fringes have been reckoned by fome to pro¬
ceed from the ordinary refraction of the Air, but with¬
out due examination of the matter. For the circuit-
fiances of the Phenomenon, fo far as I have obferved
them-, are as follows,

Q q

OBS,

[iH]

O B S. I.

I made in a piece of Lead a fmall Hole with a Pin,
whofe breadth was the 41th part of an Inch. For 21
of thofe Pins laid together took up the breadth of half
an Inch. Through this Hole I let into my darkened
Chamber a beam of the Sun's Light, and found that the
fhadows of Hairs,Thred,Pins,Straws, and fuch like lien-
der fubftances place! in this beam of Light, were confider-
ably broader than they ought to be, if the rays of Light
palled on by thefe Bodies in right Lines. And particu¬
larly a Hair oi a Man’s Head, whofe breadth was but
the 280th part of an Inch, being held in this Light, at
the diftance of about twelve Feet from the Hole, did
eaft a fhadow which at the diftance of four Inches from
the Hair was the fixtieth part of an Inch broad, that is,
above four times broader than the Hair, and at the di¬
ftance of two Feet from the Hair was about the eight
and twentieth part of an Inch broad, that is, ten times
broader than the Hair, and at the diftance of ten Feet
was the eighth part of an Inch broad, that is 3 5 times
broader.

Nor is it material whether the Hair be incompaffed
with Air, or with any other pellucid fubftance. For I
wetted a poliftied plate of Glais, and laid the Hair in
the Water upon the Glafs, and then laying another po-
lifhed plate of Glafs upon it, fo that the Water might
fill up the fpace between the GlaiTes, I held theni in
the aforefaid beam of Light, lb that the Light might
pais through them perpendicularly, and the fhadow
of the Hair was at the lame diftances as big as before.

The

C»$]

The fhadows of fcratches made in poliflied plates of
Glafs were alfo much broader than they ought to be,
and the Veins in poliffied plates of Glafs did alfo call the
like broad fhadows. And therefore the great breadth
of thefe fhadows proceeds from feme other caufe than
the refraction of the Air.

Let the Circle X reprefent the middle of the Hair; pin\
A D G, B E H, C F I, three rays paffing by one fide of
the Hair at feveral diftances; KNQ, LOR, MPS,
three other rays paffing by the other fide of the Hair at
the like diftances; D, E, F and N, Q, P, the places
where the rays are bent in their paffage by the Hair;

G, H, I and Q, R, S, the places where the rays fall on
a Paper GQ; IS the breadth of the ffiadow of the Hair
caft on the Paper, and T I, V S, two rays paffing to the
points I and S without bending when the Hair is taken
away. And it’s manifeft that all the Light between
thefe two rays AI and V S is bent in paffing by the
Hair, and turned afide from the ffiadow IS, becaufe if
any part of this Light were not bent it would fall on
the Paper within the ffiadow, and there illuminate the
Paper contrary to experience. And becaufe when the
Paper is at a great diftance from the Hair, the ffiadow
is broad, and therefore the rays TI and VS are at a
great diftance from one another, it follows that the
Hair a£ts upon the rays of Light at a good diftance in
their paffing by it. But the aftion is ftrongeft on the
rays which pafs by at leaft diftances, and grows weaker
and weaker accordingly as the rays pafs by at diftances
greater and greater, as is reprefented in the Scheme:

For thence it comes to pals, that the ffiadow of the
Hair is much broader in proportion to the diftance of

(2 q a the

[nd]

the Paper from the Hair, when the Paper is nearer the
Hair than when it is at a great diftance from it.

O B S. II.

* H

The ffiadows of all Bodies ( Metals, Stones, Glafs,
Wood, Horn, Ice, toV. j in this Light were bordered
with three parallel fringes or bands of coloured Light,
whereof that which was contiguous to the lhadow was
broadeft and mod luminous, and that which was re-
moteft from it was narroweft, and fo faint, as not eafily
to be vifible. It was difficult to diftinguiffi the Colours
unlefs when the Light fell very obliquely upon a fmooth
Paper, or fome other fmooth white Body, fo as to make
them appear much broader than they would otherwife
do. And then the Colours were plainly vifible in this
order: The firft or innermoft fringe was violet and deep
blue next the ffiadow, and then light blue, green and
yellow in the middle, and red without. The fecond
fringe was almoft contiguous to the firft, and the third
to the fecond, and both were blue within and yellow
and red without, but their Colours were very faint
efpecially thofe of the third. The Colours therefore
proceeded in this order from the ffiadow, violet, indico,
pale blue, green, yellow, red; blue, yellow, red; pale
blue, pale yellow and red. The fhadows made by
fcratches and bubbles in poliflied plates of Glafs were
bordered with the like fringes of coloured Light. And
if plates of Looking-glafs floop’d off near the edges with
a Diamond cut, be held in the fame beam of Light, the
Light which paffes through the parallel planes of the
Glafs will be be bordered with the like fringes of Co¬
lours

[ 11 7 3

lours where thofe Planes meet with the Diamond cut,
and by this means there will fometimes appear four or
five fringes of Colours. Let A B, C D reprefent the Fig
parallel planes of a Looking-glafs, and BD the plane
of the Diamond-cut, making at B a very obtufe Angle
with the plane A B. And let all the Light between the
rays E NI and F B M pafs diredtly through the parallel
planes of theGlafs, and fall upon the Paper between I
and M, and all the Light between the rays GO and
HD be refracted by the oblique plane of the Diamond
cut B D,and fall upon the Paper between K and L ; and
the Light which paffes direftly through the parallel
planes of the Glafs, and falls upon the Paper between
I and M, will be bordered with three or more fringes
at M.

O B S. III.

When the Hair was twelve Feet diftant from the 1
Hole, and its fiiadow fell obliquely upon a flat white
fcale of Inches and parts of an Inch placed half a Foot
beyond it, and alfo when the fliadow fell perpendicu¬
larly upon the fame fcale placed nine Feet beyond it;
I meafured the breadth of the lhadow and fringes as
accurately as I could, and found them in parts of an
Inch as follows*

[n8]

At the diftance of ^oot^ *Feet

The breadth of the Shadow

S 4

The breadth between the middles of the
brighteft Light of the innermoft fringes
on either fide the fhadow

T — or J -

3 * 3 *

7 .

S°

The breadth between the middles of the
brighteft Light of the middlemoft frin¬
ges on either fide the lhadow

~ I

237

l T

The breadth between the middles of the
brighteft Light of the outmoft fringes
on either fide the fhadow

s ° r 78 ]

The diftance between the middles of the
brighteft Light of the fir ft and fecond
fringes

120

The diftance between the middles of the
brighteft Light of the fecond and third
fringes

jr^

17a

The breadth of the luminous part (green,
white, yellow and red ) of the firft
fringe

173

The breadth of the darker fpace between
the firft and fecond fringes.

4 ?

The breadth of the luminous part of the
fecond fringe

290

I__

The breadth of the darker fpace between
the fecond and third fringes.

340

03 |

Thcfe

[ up]

Thefe meafures I took by letting the Ihadow of the
Hair at half a Foot diftance fall lb obliquely on the
fcale as to appear twelve times broader than when it
fell perpendicularly on it at the fame diftance, and bet¬
ting down in this Table the twelfth part of the mea¬
fures I then took.

OBS. IV.

When the fhadovv and fringes were caft obliquely
upon a fmooth white Body, and that Body was remo¬
ved further and further from the Hair, the firft fringe
began to appear and look brighter than the reft of the
Light at the diftance of lefs than, a quarter of an Inch
from the Hair, and the dark line or Ihadow between
that and the fecond fringe began to appear at a lefs di¬
ftance from the Hair than that of the third part of an
Inch* The fecond fringe began to appear at a diftance
from the Hair of lefs than half an Inch, and the Ihadow
between that and the third fringe at a diftance lefs than
an Inch, and the third fringe at a diftance lefs than three
Inches. At greater diftances they became much more
fenfible, but kept very nearly the fame proportion of
their breadths and intervals which they had at their firft
appearing. For the diftance between the middle of the
firft and middle of the fecond fringe, was to the diftance
between the middle of the fecond and middle of the
third fringe, as three to two, or ten to feven. And
the laft of thefe two diftances was equal to the breadth
of the bright Light or luminous part of the firft fringe.
And this breadth was to the breadth of the bright Light
of the fecond fringe as feven to four, and to the dark

interval.

[II2]

interval of the firft and fecond fringe as three to two,
and to the like dark interval between the fecond and
third as two to one. For the breadths of the fringes
feemed to be in the progreffion of the numbers i, i/^ ,
f/ \ and their intervals to be in the fame progreffion
with them ; that is, the fringes and their intervals to¬
gether to be in the continual progreffion of the numbers
i, / {? f /~, f/ 1 -^ f/] > or thereabouts. And thefe pro¬
portions held the fame very nearly at all diftances from
the Hair 5 the dark Intervals of the fringes being as
broad in proportion to the fringes at their firft appea¬
rance as afterwards at great diftances from the Hair,
though not ib dark and diftinft.

OBS, V.

The Sun fhining into my darkened Chamber through
a Hole a quarter of an Inch broad ; I placed at the di-
ftance of two or three Feet from the Hole a Sheet of
Paft-board, which was black’d all over on both fides,
and in the middle of it had a Hole about three quarters
of an Inch fquare for the Light to pafs through. And
behind the Hole I fattened to the Paft-board with Pitch
the blade of a fliarp Knife, to intercept fome part of
the Light which patted through the Hole. The planes
of the PafTboard and blade of the Knife were parallel
to one another, and perpendicular to the rays. And
when they were fo placed that none of the Sun’s Light
fell on the Paft-board, but all of it patted through the
Hole to the Knife, and there part of it fell upon the
blade of the Knife, and part of it palled by its edge:
I let this part of the Light which patted by, fall on a

white

[ 121 ]

white Paper two or three Feet beyond the Knife, and
there faw two ftreams of faint Light fhoot out both
ways from the beam of Light into the ftiadow like the
tails of Comets. But becaufe the Sun’s direct Light by
its brightnefs upon the Paper obfcured thefe faint
ftreams, fo that I could fcarce fee them, 1 made a little
Hole in the midft of the Paper for that Light to pafs
through and fall on a black cloth behind it; and then
I law the two ftreams plainly. They were like one
another, and pretty nearly equal in length and breadth
and quantity of Light. Their Light at that end nex
the Sun’s direft Light was pretty ftrong for the fpace o
about a quarter of an Inch, or half an Inch, and in al
its progrefs from that diredt Light decreafed gradually
till it became infenfible. The whole length of either o
thefe ftreams meafured upon the Paper at the diftano
of three Feet from the Knife was about fix or eigh
Inches; fo that it fubtended an Angle at the edge of
the Knife of about ioorn, dr at moft 14 degrees.
Yet fometimes I thought I faw it (hoot three or four
degrees further, but with a Light fo very faint that I
could fcarce perceive it, and fulpefted it might ( in
fome meafure at leaft) arile from fome other caufe than
the two ftreams did. For placing my Eye in that Light
beyond the end of that ftream which was behind the
Knife, and looking towards the Knife, I could fee a
line of Light upon its edge, and that not only when
my Eye was in the line of the ftreams, but alfo when
it was without that line either towards the point of the
Knife, or towards the handle. This line of Light ap¬
peared contiguous to the edge of the Knife, and was
narrower than the Light of the innermoft fringe, and

R r narrowed

[ 122 ]

narroweft when my Eye was furtheft from the direct
Light, and therefore feemed to pafs between the Light
of that fringe and the edge of the Knife, and that
which paffed neareft the edge to be mo ft bent, though
not all of it.

O B S. VI

I placed another Knife by this fo that their edges
might be parallel and look towards one another, and
that the beam of Light might fall upon both the Knives,
and fome part of it pafs between their edges. And
when the diftance of their edges was about the 400th
part of an Inch the ftream parted in the middle, and
left a lhadow between the two parts. This fhadow
was fo black and dark that all the Light which paffed
between the Knives feemed to be bent, and turned afide
to the one hand or to the other. And as the Knives ftill
approached one another the fhadow grew broader, and
the ftreams fhorter at their inward ends which were
next the lhadow, until upon the con tad: of the Knives
the whole Light vanilhed leaving its place to the
lhadow.

And hence I gather that the Light which is leaffc
bent, and goes to the inward ends of the ftreams, paf-
fes by the edges of the Knives at the greateft diftance,
and this diftance when the fhadow begins to appear be¬
tween the ftreams is about the eight-hundredth part of
an Inch. And the Light which paffes by the edges of
the Knives at diftances ftill lefs and lets is more and
more bent, and goes to thofe parts of the ftreams which
are further and further from the dired Light, becaufe

when

r 12 3 i

when the Knives approach one another till they touchy
thole parts of the ftreams vanilh laft which are furtheft
from the direct Light*

O B S. VIL

In the fifth Obfervation the fringes did not appear,
but by reafon of the breadth of the Hole in the Win¬
dow became fo broad as to run into one another, and
by joyning make one continued Light in the beginning
of the ftreams. But in the fixth, as the Knives ap¬
proached one another, a little before the Shadow ap¬
peared between the two ftreams, the fringes began to
appear on the inner ends of the ftreams on either fide
of the direct Light, three on one fide made by the edge
of one Knife, and three on the other fide made by the
edge of the other Knife. They were diftinfteft when
the Knives were placed at the greateft diftance from the
Hole in the Window, and ftill became more diftinft by
making the Hole lefs, infomuch that I could fometimes
fee a faint lineament of a fourth fringe beyond the three
above-mentioned. And as the Knives continually ap¬
proached one another, the fringes grew diftinTer and
larger until they vanilhed. The outmoft fringe va¬
nished firft, and the middlemoft next, and the inner-
moft laft. And after they were all vanilhed, and the
line of Light which was in the middle between them
was grown very broad, enlarging it fielf on both fides
into the ftreams of Light defcribed in the fifth Obfer¬
vation, the above-mentioned fhadow began to appear
in the middle of this line, and divide it along the middle
into two lines of Light, and increafed until the whole

Rr 2 Light

. . . . t*H]

Light vanifhed. This inlargement of the fringes was
fo great that the rays which go to the innermoft fringe
feemed to be bent above twenty times more when this
fringe was ready to vanilh, than when one of the Knives
was taken away.

And from this and the former Obfervation compared,
I gather, that the Light of the firft fringe paflfed by the
edge of the Knife at a diftance greater than the eight-
hundredth part of an Inch, and the Light of the fecond
fringe palled by the edge of the Knife at a greater di¬
ftance than the Light of the firft fringe did, and that
of the third at a greater diftance than that of the fe¬
cond, and that of the ftreams of Light defcribed in
the fifth and fixth Obfervations palled by the edges
of the Knives at lefs diftances than that of any of the
fringes..

o b a vin.

I caufed the edges of two Knives to be ground truly
ftreight, and pricking their points into a board fo that
their edges might look towards one another, and meet¬
ing near their points contain a rectilinear Angle, I fafl>
ned their handles together with Pitch to make this
Angle invariable. The diftance of the edges of the
ATnives from one another at the diftance of four Inches
from the angular point, where the edges of the Knives
met, was the eighth part of an Inch, and therefore the
Angle contained by the edges was about i degr. 54/.
The Knives thus fixed together I placed in a beam of
the Sun's Light, let into my darkened Chamber through
a Hole the 42th part of an Inch wide, at the diftance

[ 12 $ ]

of ten or fifteen Feet from the Hole, and let the Light
which pafled between their edges fall very obliquely
upon a lmooth white Ruler at the diftance of half an
Inch, or an Inch from the Knives, and there faw the
fringes made by the two edges of the Knives run along
the edges of the lhadows of the Knives in lines parallel
to thole edges without growing fenfibly broader, till
they met in Angles equal to the Angle contained by the
edges of the Knives, and where they met and joyned
they ended without eroding one another. But if the
Ruler was held at a much greater diftance from the
Paper, the fringes became fomething broader and. broader
as they approached one another, and after they met
they crofted one another, and then became much broader
than before.

Whence I gather that the diftances at which the
fringes pafs by the Knives are not increafed nor altered
by the approach of the Knives, but the Angles in which
the rays are there bent are much increafed by that ap¬
proach ; and that the Knife which is neareft any ray
determines which way the ray lhall be bent, and the
other Knife increafes the bent.

O B S.. IX,.

When the rays fell very obliquely upon the Ruler at
the diftance of the third part of an Inch from the Knives,
the dark line between the firft and fecond fringe of the
lhadow of one Knife, and the dark line between the
firft and fecond fringe of the lhadow of the other Knife
met with one another, at the diftance of the fifth part
of an Inch from the end of the Light which pafled be¬
tween.

tween the Knives at the concourfe of their edges. And
therefore the diftance of the edges of the Knives at the
meeting of thefe dark lines was the i6othpart of an
Inch, For as four Inches to the eighth part of an Inch,
fo is any length of the edges of the Knives rneafured
from the point of their concourfe to the diftance of the
edges of the Knives at the end of that length, and fo is
the fifth part of an Inch to the 16oth part. So then the
dark lines above-mentioned meet in the middle of the
Light which pafles between the Knives where they are
diftant the 16oth part of an Inch, and the one half of
that Light pafles by the edge of one Knife at a diftance
not greater than the 320th part of an Inch, and falling
upon the Paper makes the fringes of the lhadow of that
Knife, and the other half pafles by the edge of the
other Knife, at a diftance not greater than the 320th
part of an Inch, and falling upon the Paper makes the
fringes of the lhadow of the other Knife. But if the
Paper be held at a diftance from the Knives greater than
the third part of an Inch, the dark lines above-men¬
tioned meet at a greater diftance than the fifth part of
an Inch from the end of the Light which pafled be¬
tween the Knives at the concourfe of their edges; and
therefore the Light which falls upon the Paper where
thofe dark lines meet pafles between the Knives
where their edges are diftant above the 160th part of
an Inch,

For at another time when the two Knives were di¬
ftant eight Feet and five Inches from the little Hole in
the Window, made with a fmall Pin as above, the Light
which fell upon the Paper where the aforefaid dark
lines met. pafled between the Knives, where the di~

ftance

[ 127 ]

fiance between their edges was as in the following
Table, when the diftance of the Paper from the Knives
was alio as follows*

Difiances of the Paper
from the Kjvives in
Inches .

Difiances between the edges
of the }(jiives in mtlle-
fimal parts of an Inch .

ij.

o’oi 2.

3 ?

ooao.

8t-

q’o’ 34 .

32.

O'C^y,

96,

o’o8i.

0^087.

And hence I gather that the Light which makes the
fringes upon the Paper is not the fame Light at all di-
fiances of the Paper from the Knives, but when the Pa¬
per is held near the Knives, the fringes are made by
Light which paffes by the edges of the Knives at a lefs
diftance, and is more bent than when the Paper is held
at a greater diftance from the Knives,

O B S* X.

When the fringes of the lhadows. of the Knives fell
perpendicularly upon a Paper at a great diftance from
the Knives,- they were in the form of Hyperbolas, and
their dimenfions were as follows* LetCA, CB repre-
fent lines drawn upon the Paper parallel to the edges of
the Knives, and between which all the Light would
fall, if it paffed between the edges of the Knives with¬
out inflexion; DE a right line drawn throughC making

[ 128 ]

the Angles A CD, BCE, equal to one another, and
terminating all the Light whith falls upon the Paper from
the point where the edges of the Knives meet; eis, fkt,
and glv, three hyperbolical lines reprefenting the ter¬
minus of the fhadow of one of the Knives, the dark line
between the firft and fecond fringes of that lhadow, and
the dark line between the fecond and third fringes of
the fame fhadow ; x i p, y k q and z 1 r, three other Hy¬
perbolical lines reprefenting the terminus of the fhadow
of the other Knife, the dark line between the firft and
fecond fringes of that fhadow, and the dark line be¬
tween the fecond and third fringes of the fame fhadow.
And conceive that thefe three Hyperbolas are like and
equal to the former three, and crofs them in the points
i, k and 1, and that the fhadows of the knives are termi¬
nated and diftinguifhed from the firft luminous fringes
by the lines eis and xip, until the meeting and crof¬
ting of the fringes, and then thofe lines crofs the fringes
in the form of dark lines, terminating the firft luminous
fringes within fide, and diftinguifhing them from ano¬
ther Light which begins to appear at i, and illuminates
all the triangular fpace ipDEs comprehended by thefe
dark lines, and the right line DE. Of thefe Hy¬
perbolas one Afymptote is the line D E, and their other
Afymptotes are parallel to the lines C A and C B. Let
r v reprefent a line drawn any where upon the Paper
parallel to the Afymptote D E, and let this line crofs
the right lines A C in m and B C in n, and the fix dark
hyperbolical lines in p, q, r; s, t, v ; and by meafuring
the diftances ps, qt, rv, and thence colleiting the
the lengths of the ordinates n p, n q, n r or m s, m t,
mv, and doing this at feveral diftances of the line rv,

[ I2p]

from the Afymptote DE you may find as many points
of thefe Hyperbolas as you pleafe, and thereby know
that thefe curve lines are Hyperbolas differing little from
the conical Hyperbola. And by meafuring the lines
C i, C k , Cl, you may find other points of thefe
Curves.

For inftance, when the Knives were diftant from the
Hole in the Window ten Feet, and the Paper from the
Knives 9 Feet, and the Angle contained by the edges of
the Knives to which the Angle ACB is equal,was fut>
tended by a chord which was to the Radius as 1 to 52,
and the diftance of the line rv from the Afymptote DE
was half an Inch : I meafured the lines ps, qt, rv,
and found them 0^5, 0*65, 0*98 Inches refpedivefy,
and by adding to their halfs the line [ mn (which here
was the 128th part of an Inch, or o’ooyS Inches ) the
fums np, nq, nr, were o ? i8i8, 0^318, 0^978 In¬
ches. I meafured alfo the difiances of the brightefi
parts of the fringes which run between pqarid st, qr
and tv, and next beyond r and v, and found them o’j,
o’8, and 1 ’ 17 Inches,

OBS. XL

The Sun fhihing into my darkened Room through a
finall round Hole made in a plate of Lead with a llender
Pin as above ; I placed at the Hole a Pilfer to refrad
the Light, and form on the oppofite Wall theSpedrum
of Colours, defcribed in the third Experiment of the
firftBook. And then I found that the ihadows of all
Bodies held in the coloured Light between the Pf life
and the Wall, were bordered with fringes of the Cdlotif

S s ‘ J of

[ 13 ° 3

of that Light in which they were held. In the full red
Light they were totally red without any fenfible blue
or violet, and in the deep blue Light they were totally
blue without any fenfible red or yellow ; and fo in the
green Light they were totally green, excepting a little
yellow and blue, which were mixed in the green Light
of the Prifm. And comparing the fringes made in the
feveral coloured Lights, I found that thofe made in the
red Light were largeft, thofe made in the violet were
leaf!:, and thofe made in the green were of a middle
bignefs. For the fringes with which the fhadow of a *
Man’s Hair were bordered, being meafured crofs the
fhadow at the diftance of fix Inches from the Hair ; the
difiance between the middle and moft luminous part of
the firft or innermoft fringe on one fide of the fhadow,
and that of the like fringe on the other fide of the lha-
dow, was in the full red Light ^ of an Inch, and in
the full violet tg. And the like diftance between the
middle and moft luminous parts of the fecond fringes on
either fide the fhadow was in the full red Light h, and
in the violet • of an Inch. And thefe diftances of the
fringes held the fame proportion at all diftances from
the Hair without any fenfible variation.

So then the rays which made thefe fringes in the red
Light pafled by the Hair at a greater diftance than thofe
did which made the like fringes in the violet; and there¬
fore the Hair in caufing thefe fringes adted alike upon
the red Light or leafi refrangible rays at a greater di¬
ftance, and upon the violet or moft refrangible rays at
a lefs diftance, and by thofe actions difpofed the red
Light into larger fringes, and the violet into fmaller,
and the Lights of intermediate Colours into fringes of

inter-

Cl 3 1 3

intermediate bigneffes without changing the Colour of
of any fort of Light,

When therefore the Hair in the fir ft and fecond of
thefie Oblervations was held in the white beam of the
Sun’s Light, and caft a fhadow which was bordered with
three fringes of coloured Light, thofe Colours arofe not
from any new modifications impreft upon the rays of
Light by the Hair, but only from the various inflections
whereby the feveral forts of rays were feparated from
one another, which before feparation by the mixture
of all their Colours, compofed the white beam of the
Sun’s Light, but whenever fepaaated comp ole Lights
of the feveral Colours which they are originally dilpo-
fed to exhibit. In this 13 th Obfervation, where the
Colours are feparated before the Light pafles by the
Hair, the leaft refrangible rays, which when fepara¬
ted from the reft make red, were infleCted at a greater
diftance from the Hair, fo as to make three red fringes
at a greater diftance from the middle of the fhadow of
the Hair 3 and the moft refrangible rays which when
feparated make violet, were infleCted at a lefs diftance
from the Hair, fo as to make three violet fringes at a
lefs diftance from the middle of the fhadow of the Hair.
And other rays of intermediate degrees of refrangible
lity were infleCted at intermediate diftances from the
Hair, fo as to make fringes of intermediate Colours at
intermediate diftances from the middle of the fhadow

of the Hair. And in the fecond Obfervation, where
all the Colours are mixed in the white Light which
pafles by the Hair, thefe Colours are feparated by the
various inflexions of the rays, and the fringes which
they make appear all together , and the innennolf

S § a fringes

. . - , t r 3 2 J

fringes being contiguous make one broad fringe compo-
fed of all the Colours in due order, the violet lying
on the infide of the fringe next the fhadow, the red on
the outfide furtheft from the fhadow, and the blue,
green and yellow, in the middle. And, in like man¬
ner, the middlemoft fringes of all the Colours lying in
order, and being contiguous, make another broad fringe
compofed of all the Colours; and the outmoft fringes
of all the Colours lying in ordejr, and being contiguous,
make a third broad fringe compofed of all the Colours.
Thefe are the three fringes of coloured Light with
which the lhadows of all Bodies are bordered in the fe-
cond Obfervation.

When I made the foregoing Obfervations, I defigned
to repeat moft of them with more care and exaftnefs,
and to make fome new ones for determining the man¬
ner how the rays of Light are bent in their paffage by
Bodies for making the fringes of Colours with the
dark lines between them. But I was then interrupt
ted, and cannot how think of taking thefe things into
further confederation. And iince I have not finifhed
this part of my Defign, I fhall conclude, with propo-
fing only lbme Queries in order to a further learclr to
be made by others.

.... A • ‘ "r * • f

J^uery i. Do not Bodies ad upon Light at a diftance,
and by their action bend its rays, and is not this addon
(cateris fariAm) ftrongeft at the leaft diftance ?

G>u. a. Do not the rays which differ in refrangibility
differ alfo in flexibility, and are they not by their dif¬
ferent inflexions feparated from one another, fo as
after reparation to make the Colours in the three fringes

above

C 1 331

above defcribed ? And after what manner are they in-
fleCted to make thofe fringes ?

3. Are not the rays of Light in paffing by the
edges and fides of Bodies, bent feveral times backwards
and forwards, with a motion like that of an Eel ? And
do not the three fringes of coloured Light above-men¬
tioned, arife from three fuch bendings ?

Qu. 4. Do not the rays of Light which fall upon Bo¬
dies, and are reflected or refracted, begin to bend be¬
fore they arrive at the Bodies; and are they not re¬
flected, refracted and infleCted by one and the fame
Principle, adding varioufly in various circmnftances ?

5* Do not Bodies and Light add mutually upon
one another, that is to lay, Bodies upon Light in emit¬
ting, reflecting, refraCting and inflefting it, and Light
upon Bodies for heating them, and putting their parts
into a vibrating motion wherein heat con hits ?

<%. 6. Do not black Bodies conceive heat more eafily
from Light than thofe of other Colours do, by reafon
that the Light falling on them is not reflected outwards,
but enters the Bodies, and is often reflected and re-
fradted within them, until it be {tilled and loft ?

7. Is not the ftrength and vigor of the aCtion
between Light and fulphureous Bodies obferved above,
one reafon why fulphureous Bodies take fire more
readily, and burn more vehemently, then’other Bor
dies do ?

8, Do not all fixt Bodies when heated beyond a
certain degree, emit Light and fhine, and is not this
emiflion performed by the vibrating motions of their
parts ?

C134]

«%. 9. Is not fire a Body heated fo hot as to emit
Light copioufly ? For what elfe is a red hot Iron than
fire ? And what elfe is a burning Coal than red hot
Wood ?

10. Is not flame a vapour, fume or exhalation
heated red hot, that is, fo hot as tolhine? For Bodies
do not flame without emitting a copious fume, and this
fume burns in the flame. The Ignis Fatuus is a vapour
j fhining without heat, and is there not the fame diffe¬

rence between this vapour and flame, as between rot¬
ten Wood fhining without heat and burning Coals of
fire ? In diftilling hot Spirits, if the head of the ftill be
taken off, the vapour which afcends out of the Still will
take fire at the flame of a Candle, and turn into flame,
and the flame will run along the vapour from the Candle
to the Still. Some Bodies heated by motion or fermen¬
tation, if the heat grow intenfe fume copioufly, and if
the heat be great enough the fumes will fhine and be¬
come flame. Metals in fufion do not flame for want of
a copious fume, except Spelter which fumes copioufly,
and thereby flames. All flaming Bodies, as Oyl, Tal-
low, Wax, Wood, foffil Coals, Pitch, Sulphur, by
^flaming wafte and vanifh into burning fmoke, which
fmoke, if the frame be put out, is very thick and vifible,
and fometimes fmells ftrongly, but in the flame loles
its fmell by burning, and according to the nature of the
fmoke the flame is of feveral Colours, as that of Sul¬
phur blue, that of Copper opened with Sublimate
green, that of Tallow yellow* Smoke paffing through
flame cannot but grow red hot, and red hot Imoke can
V have no other appearance than that of flame.

11.

- [!?;]

<$u. n. Do not great Bodies conferve their heat the
longeft, their parts heating one another, and may not
great denfe and fix’d Bodies, when heated beyond a
certain degree, emit Light fo copiouily, as by the emif-
fion and reaction of its Light, and the reflexions and re¬
fractions of its .rays within its pores to grow ftill hot¬
ter, till it comes to a certain period of heat, luch as is
that of the Sun ? And are not the Sun and fix’d Stars
great Earths vehemently hot, whofe heat is conferved
by the greatnefs of the Bodies, and the mutual aCtion
and reaction between them, and the Light which they
emit, and whofe parts are kept from fuming away, not
only by their fixity, but alfo by the vaft weight and
denfity of the Atmofpheres incumbent upon them, and
very ftrongly comprefling them, and condenfing the va¬
pours and exhalations which arife from them ?

£$u. i a. Do not the rays of Light in falling upon the
bottom of the Eye excite vibrations in the "Tunica re -
tina ? Which vibrations, being propagated along the
folid fibres of the optick Nerves into the Brain, caufe
the fenfe of feeing. For becaufe denfe Bodies conferve'
their heat a long time, and the denfeft Bodies conferve
their heat the longeft, the vibrations of their parts are
of a lafting nature, and therefore may be propagated
along folid fibres of uniform denfe matter to a great di-
ftance, for conveying into the Brain the impreflions
made upon all the Organs of fenfe. For that motion ,
which can continue long in one and the fame part of a
Body, can be propagated a long way from one part to
another, fuppofing the Body homogeneal, fo that the
motion may not be reflected, refraCted, interrupted or;
difordered by any unevennefs of the Body.

§>u. 13.

[i 3 6]

«%£. 13. Do not feveral fort of rays make vibrations
of feveral bigneffes, which according to their bigneffes
excite fenfations of feveral Colours, much after the
manner that the vibrations of the Air, according to their
feveral bigneffes excite fenfations of feveral founds ?
And particularly do not the mo ft refrangible rays ex¬
cite the fhorteft vibrations for making* a fenfation of
deep violet, the leaft refrangible the largeft for making
a fenfation of deep red, and the feveral intermediate
forts of rays, vibrations of feveral intermediate bignef¬
fes to make fenfations of the feveral intermediate Co¬
lours ?

14. May not the harmony and difcord of Co¬
lours arife from the proportions of the vibrations propa¬
gated through the fibres of the optick Nerves into the
Brain, as the harmony and difcord of founds arifes from
the proportions of the vibrations of the Air ? For feme
Colours are agreeable, as thofe of Gold and Indico, and
others difagree*

<Qu. 15. Are not the Species of Objects feen with both
Eyes united where the optick Nerves meet before
they come into the Brain, the fibres on the right fide
of both Nerves uniting there, and after union going
thence into the Brain in the Nerve which is on the
right fide of the Head, and the fibres on the left fide
of both Nerves uniting in the fame place, and after
union going into the Brain in the Nerve which is on
the left fide of the Head, and thefe two Nerves meet¬
ing in the Brain in fuch a manner that their fibres
make but one entire Species or Picture, half of which
on the right fide of the Senforium comes from the
right fide of both Eyes through the right fide of

both

[ i37]

both optick Nerves to the place where the Nerves
meet, and from thence on the right fide of the Head
into the Brain, and the other half on the left fide of the
Senforium comes in like manner from the left fide of
both Eyes. For the optick Nerves offuch Animals as
look the fame way with both Eyes (as of Men, Dogs,
Sheep, Oxen, ) 3 c. ) meet before they come into the
Brain, but the optick Nerves of Inch Animals as do
not look the fame way with both Eyes (as of Filhes and
of the Chameleon) do not meet, if I am rightly in¬
formed.

$u. 16. When a Man in the dark preffes either cor¬
ner of his Eye with his Finger, and turns his Eye away
from his Finger, he will fee a Circle of Colours like
thofe in the Feather of a Peacock’s Tail ? Do not thefe
Colours arife from fuch motions excited in the bottom
of the Eye by the preffure of the Finger, as at other
times are excited there by Light for caufing Vifion ? And
when a Man by a ftroke upon his Eye fees a Flafii of
Light, are not the like Motions excited in the Retina
by the ftroke ?

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Book n. Plate I.

O pq ~«j ^ J ^

Book HI.Plate I.

TER Til OR DIN IS.

[" 39 ]

ENUME RATIO

LINEARUM

TERTII ORDINIS,

L Ineae Geometric^ fecundum numerum dimen- i:

fionum aequationis qua relatio inter Ordinatas Lin * < & um
& Abfciflas definitur, vel (quod perinde eft) fecum m * 5 '
dum numerum pun&orum in quibus a linea re£la
fecari poffunt^ optime diftinguuntur in Ordines*

Qua ratione linea primi Ordinis erit Re£ta fola, eas
fecund! five quadratici ordinis erunt feftiones Conicae
& Circulus, & ex tertii five cubic! Ordinis Parabola
Cubica, Parabola Neiliana, Ciffois veterum & relL
quae quas hie enumerare fufeepimus. Curva autem
primi generis, (fiquidem refta inter Curvas non eft
numeranda) eadem eft cum Linea fecundi Ordinis 5
& Curva fecundi generis eadem cum Linea Ordinis
tertii. Et Linea Ordinis infinitefimi ea eft quam
re£ta in punftis infinitis fecare poteft, qualis eft Spi¬
ralis, Cyclois, Quadratrix & linea omnis quae per
radii vel rota: revofutiones infinitas genera tur.

Tt i

SeCtionum

II.

Proprieties Se-
iHonem Conica -
rum competunt
Purvis fuperiorum
generum .

III.

Cur varum fe-
cundi generis Or~
din at a. Diame¬
tric VvrticesJCen -
tra 5 Axes,

Seftionum Conicarum proprietates praecipuae a
Geometris paffim traduntur*. Et confimiles funt pro-
prietates Curvarum fecundi generis & reliquarutn, ut
ex fequenti proprietatum praecipuarum enumera-
tione conftabit.

Nam fi redae plures parallels & ad conieam fe-
dionem utrinq; terminate ducantur, reda duas ea-
fum bifeeans bifecablt alias omnes,icteoq; dicitur D ia~
meter figmx&L redae bifedae dkuntur Ordinatim
die attend Biametrum, & concurfus omnium Dia-
metrorum eft Centrum figprae, & interfiedio Curvae &
diametri Vertex nominatur, & diameter ilia Axis
eft cui ordinatim applicates infiftunt ad angulos re¬
dos. Et ad eundem moduin in Cur vis fecundi ge¬
neris,' fi redae duae quaevis parallelae ducantur occur-
rentes Curvae in tribus pundis: reda quae ita fecat
has parallels^ ut fumma duarum partium ex uno Te-
eantis latere adcurvam terminatarum aequeturparti
tertiae ex altero latere ad curvam terminates, eodem
inodo fecabit omnes alias his parallelas curvaeq; in
tribus pundis occurrentes redas, hoc eft,,ita ut turn-
in a partium duarum ex uno ipfius latere femper
aequetur parti tertiae ex altero latere. Has itaq; tres
partes quae hinc inde aequantur, Ordinatim apflu
eatevs & redam fecantem cui ordinatim applicantur
‘Diametrum. & interfedionem diametri & curve zVer~
ticem & concurfum duarum diametrorum Centrum
nominare licet. Diameter autem ad Ordinatas re-
dangula fi modo aliqua fit, etiam Axis did poteft~.
& ubi omnes diametri in eodem pundo concurrunt
iftud erit Centrum generale*

[ H 1 ]

Hyperbola primi generis duas Afymptotos , ea fo* # JV.
eundi tres,ea tertii quatuor & non plures habere po-
teft, & lie in reliquis. Et quemadmodum partes^,
lines cujufvis refits inter Hyperbolam Conicam &
duas ejus Afymptotos lunt hinc inde squales : lie irj(

Hyperbolis fecundi generis fi ducatur reda qusvi/s
fecans tam Curvam quam tres ejus Afymptotos 4n
tribus pundis, fumma duarum partium iftius rdfts
qus a duobus quibufvis Alymptptis in eandem pla-
gam ad duo punda Curvs extenduntur squali§ erit
parti tertis quae a tertia Afymptoto in plagam con-
trariam ad tertium Curvae pundum extenditur;

Et quemadmodum in Conicis fedionibus non Pa» v.
rabolicis quadratum Ordinatim applicate, hoc eft ,^ LaterareB4m-
redangulum Ordinatarum quae ad contrarias par- 4 ^ ^ ^
tes Diametri ducuntur, eft ad redangulum partium
Diametri quae ad Vertices Ellipfeos vel Hyperbolae
terminantur,ut data quaedam linea. quae dicitur Lam s
reSlim^ ad partem diametri quae inter Vertices jacet
& dicitur Lattes tranfverfum : fic in Curvis non Para-
bolicis fecundi generis Parallelepipedum fub tribus
Ordinatim applicatis eft ad Parallelepipedum fub par¬
ti bus Diametri ad Ordinatas & tres Vertices figure ab-
feiffis, in ratione quadam data : in qua ratione ft fix-
mantur tres reds ad tres partes diametri inter ver¬
tices figuts fitas lingulae ad lingulas, tunc ills tres
reds did poffunt Late?'a reMa figurs, & ills partes.-
Diametri inter Vertices Later a tranfverfa .. Et ficufc
in Parabola Conica qus adunam & eandem diarne-
trum unicum tantum habet Verticem, redangulum;
fub Ordinatis squatur redangulofub parte Diametri
qus ad Ordinatas & Verticem abfeinditur Sereda

quadam

[ H 2 ]

quadam data qux Latus return dicitmqfic in Curvis
fecundi generis qux non nili duos habent Vertices ad
eandem Diametrunq Parallelepipedum fub Ordinatis
tribus xquatur Parallelepipedo fub duabus partibus
Diametri ad Ordinatas & Vertices illos duos abfciffis^
& reda quadam data qux proinde Latm reclum
dici poteft.

vi* Deniq; iicut in Conicis fedionibus ubi dux paral-

rum fub Paraiie- lelee ad murvam utrmq; terminats iecantur a dua-
Urumfegmemis. bus parallelis ad Curvam utrinq; terminatis, prima

a tertia & fecunda a quarta, redangulum partium
prims eft ad redangulum partium tertia ut redan-
gulum partium fecund® ad redangulum partium
quart®: lie ubi quatuor tales reds occurrunt Curva
fecundi generis lingula in tribus pundis, paralleled
pipedum partium prima reda erit ad parallelepide-
dum partium tertia, ut parallelepipedum partium
fecunda ad parallelepipedum partium quarta.

„ V!I - ^ Curvarum fecundi & fuperiorum generum aque
boiica&Pa/abo-atq; prmu crura omnia m mhmtum progredientia
Uca & eorum pia- vel Hyper Mici funt generis vel TaraMtci. Crus Hy~
per Miami voco quod ad Afymptoton aliquam in in¬
finitum appropinquat ^TaraMicum quod Afymptoto
deftituitur. Hxc crura ex tangentibus optime dig-
nofeuntur. Nam fi pundum contadus in infinitum
abeat tangens cruris Hyperbolici cum Afymptoto
coincidet & tangens cruris Parabolici in infinitum
recede^ evanefcet & nullibi reperietur. Invenitur
igitur Afymptotos cruris cujufvis quxrendo tangen-
tem cruris illius ad pundum infinite diftans. Plaga
autem cruris infiniti invenitur quxrendo pofitionem
redx cujufvis qux tangenti parallela. eit ubi pun¬
dum

''wins*

C H3l

Sum conta&us in infinitum abit. Nam haec recta
in eandem plagam cum crure infinite dirigitur.

Linear omnes Ordinis primi, tertii, quintq fep- VIIL
timi & imparis cujufq; duo habent ad minimum
crura in infinitum verfus plagas oppofitas progre- generis fscundi ad
dientia. Et lineae omnes tertii Ordinis duo habent ^ uatl0num ^4^
ejuimodi crura m plagas oppohtas progredientia m pri?
quas nulla alia earum crura infinita (praeterquam
in Parabola Cartefiana) tenduntv Si crura ilia
lint Hyperbolici generis., fit GAS eorum Afymp-
totos & huic parallela agatur reSta quamvis C B c
ad Curvam utrinque ( fi fieri poteft ) tenninata
eademq; bifecetur in pundto X, & locus pundti il- F &
lius X erit Hyperbola Conica ( puta X $ ) cujus
una Afymptotos eft AS. Sit ejus altera Afymp-
totos A B, & aaquatio qua relatio inter Ordinatam
BC & Abfciflam AB definituig fi AB dicatur x &

B C y 5 Temper induet hanc formam x y y -|- e y = a x 5
-|-bxx-(-cx~|--d. Ubi termini e, a 5 b^ c 5 d 5 defig-
nant quantitates datas cum fignis fuis -\- & —< affe-
<ftas 0 quarum quaelibet deeffe poffuntmodo ex earum
deledtu figura in fedionem conicam non vertatur. -
Poteft autem Hyberbola ilia Conica cum afympto-
tis fuis coin cider e, id eft pundum X in reda AB,
locari: & tunc terminus-j-ey deeft.

At fi reda illaCB c non poteft utrinq; ad Curvam
terminari fed Curvae in unico tan turn pundo occur-
lit: age quamvis pofitione datamredam AB afymp-
toto AS occurrentem in A, ut & aliam quamvis BC
afymptoto illi parallelam Curvaeque occurrentem in
pundo C 5 & asquatio qua relatio inter Ordinatam

IX.

Cafm fecund'mj

Cafu-s tertim.

■Cafm quartm.

XII.

- Nomi/m forma-

rum.

. . C144 ]

B C Sc AbfcilTam A B definitur, Temper induet hanc
formam x y = a x* b x x-|- c x d.

Quod fi crura ilia oppollta Parabolici lint generis,
recta C B c ad Curvam utrinque, fi fieri poteft, ter-
minata in plagam cruruni ducatur & bifecetur in B,
& locus pundi B erit linea reda. Sit ilia AB, ter-
minata ad datum quodvis pundum A, & oequatio
qua relatio inter Ordinatam BC & AbfcilTam AB
definitur, temper induet hanc formam, yy = ax s
-pbxx-Ecx-j- d.

At vero fi reda ilia C B c in unico tantum pundo
occurrat Cufvse, ideoq; ad Curvam utrinq; terminari
non poffit: fit pundum illud C, & incidat reda ilia
ad pundum B in redam quamvis aliam pofitione
datam & ad datum quodvis pundum A terminatam
A B : Sc aequatio qua relatio inter Ordinatam B C Sc
AbfcilTam AC definitur femper induet hanc formam,
y = ax 5 -fbxx^cx -f d.

Enumerando curvas horum cafuum, Hyperbolam
vocabimus injcnptam quae tota jacet in Afymptotoii
angulo ad inftar Hyperbola conicae, circumfcnptam
qua? Afymptotos fecat Sc partes abtciffas in finu fuo
ampleditur, ambigenam quae uno crure infinito in-
fcribitur Sc altero circumfcribitur, convergentem
eujus crura concavitate fua feinvicem refpiciunt Sc
in plagam eandem dir iguntur ^divergentem eujus crura
convexitate fua feinvicem recipiunt 8c in plagas con¬
traries diriguntur, cruribus contrariis preedit am eujus
crura in partes contrarias convexa funt 8c in plagas
contrarias infinita, Conchoidalem quae vertice concavo
& cruribus divergentibus ad alymptoton applicatur,
dngiiimam quae flexibus contrariis afymptoton fecat

C r 4$ ]

8c utrinq; in crura contra ria producitur, cruciformem
qua conjugatam decuffat, nodatam qus feipfam de-
cuffat in orbem redeundo, cufpidatam cujus partes
dux in angulo contadus concurrunt & ibi terminan-
tur, punttatam qure conjugatam habet Ovalem infi¬
nite parvam id eft pundum, Sc fur am quse per im-
poffibilitatem duarum radicum Ovali, Nodo, Cuf-
pide Sc Pundo conjugato privatur. Eodem fenfu
Parabolam quoq; convergeniem , divergentem , cruri-
bus contrarm pneditam , cruciformem , nodatam , cuf-
fidatamj punilatam 8c puram nominabimus.

In cai'u primo li terminus a x* affirmativus eft Fi- XIIL , ,
gura erit Hyperbola triplex^ cum fex cruriblxs Hy- redunimu&
perbolicis qusjuxta tres Alymptotos quarum nulls tribm ct-
font parallels in infinitum progrediuntur,binge juxta
unamquamq; in plagas contrarias. Et hx Alymp-
toti fi terminus b x x non deeft fe mutuo fecabunt
in tribus pundis triangulum (Dd^J inter fe con-
tinentes, fin terminus bxx deeft convergent omnes
ad ,idem pundum. In priori cafu cape A D =

& Ad = Ac/' = ~~ rs ac junge Dd, D^, 8c erunt
AD, Dd, D^tres Afymptoti. In pofteriori due
ordinatam quamvis B C, & in ea utrinq; produda
cape hine inde B F & Bf fibi mutuo sequales &
in ea ratione ad A B quam habet /d ad a, j'unpeq:

AF, Af, & erunt AB, AF, Af tres Alympoti.

Hanc autem Hyperbolam vocamus redundantem
quia numero crurum Hyperbolicorum Sediones Co¬
rneas foperat.

In Hyperbola omni redundante fi neq; terminus ^ XIV ’-

e } defit neq, i n oo — qac squale ft ae• y a curva nul -perboUdiametric
lam habebit diametrum, fin eorum alterutrum ac- & (itu crurum

- - infinitorum*

cidat

.[■4<S]

cidat curva habebi# unicam diametrum., & tres E
utrumque. Diameter autem femper tranftt per in-
terfedionem duarum Afymptoton & bilecat redas
omnes quae ad Afymptotos illas utrinq; terminantur
& parallels funt & Afymptoto tertiae. Eftq; abfciffa
A B diameter Figurae quoties terminus ey deeft.
Diametrum vero abfolute didam hie & in fequen-
tibus in vulgari fignificatu ufurpo, nempe pro ab¬
fciffa quae paffim habet ordinatas binas aequales ad
idem pundum hinc inde infiftentes.
xv. Si Hyperbola redundans nullam habet diametrum

^emredmdantes eperantur iEquationis hujus a X 4 +b X* A c X X A d X
qu*diametrode- A 4 = ° radices quatuor feu valores ipfius x. Eae

: i l Znt U Afydjtl funt0 A P 5 A w » A *, A P- Erigantur ordinate
ios triangidum PT, WT •) Trfj p tj & hae tangent Curvam in pundis
ententes. totidem T, t, it, & tangendo dabunt limites Cur-

vx per quos fpecies ejus innotefeet.
v- Nam ft radices omnes AP 5 A*®*, A^q Ap funt

reales., ejufdein figni & inaequales, Curva conftat ex
tribus Hyperbolis ., ( infer,ipta circumfcripta & am-
bigena ) cum Ovalt. Hyperbolarum una jacet ver-
fus D 1 altera verfus d, tertia verlus ^ & Ovalis
femper jacet intra triangulam Dd d \ atq; etiam in¬
ter medios limites n & r 7 in quibus utiq; tangitur
ab ordinatis nfi & ot. Et haec eft fpecies prima.

3,4* Si e radicibus duae maxima A ^ A vel duae mi¬

nima AP, A^ aequantur inter fe, & ejufdem funt
figni cum alteris duobus., Ovalis & Hyperbola cir¬
cumfcripta fibi inxicem junguntur coeuntibus earum
pundis contadusT & t vel T & ^ & crura Hyper¬
bolae fefe decuffando in Ovalem continual! tur 5 ftgu-
ram mdatcwi efficientia. Quae fpecies eft fecunda.

[ r 47 1

Si e radicibus tres maximae A/, A t, A*r, vel tres Fig. 5,at
minimae A^, A*r, AP oequentur inter fe 5 Nodus in
cuff idem acutiffimum convertetur. Nam crura duo
Hyperbolae circumfcriptae ibi in angulo contadus
concurrent & non ultra producentur. Et haec eft
fpecies tertia.

Si e radicibus duae media' A^ & Ax a'quentur in- 7.
ter fe, punda contadus T & 7 coincidunt, & propte-
rea Ovalis interjeda in pundum evanuit, & conftat
figura ex tribus Hyperbolis, infcripta, circumfcripta
& ambigena cum fundi0 conjugate,, Qua eft fpecies
quarta.

Si duae ex radicibus funt impoffibiles & reliquae
duae inaequales & ejufdem figni (nam figna contraria
habere nequeunt,) pnra habebuntur Hyperbola tres
fineOvali vel Nodo vel cufpide vel pundo conju¬
gate,, & hae Hyperbola vel ad latera trianguli ab
Afymptotis comprehenft vel ad angulos ejus jacebunt
& perinde fpeciem vel quintam vel fextam corifti- \\

tuent.

Si e radicibus duae funt aequales & alterae duae 9,10,15^
vel impoffibiles funt vel reales cum fignis quae a fig-
nis aequalium radicum diverfa funt, figura crucifor -
mis habebitur, nempe duae ex Hyperbolis feinvicem
decuffabunt idq; vel ad verticem trianguli ab A-
fymptotis comprehenfi, vel ad ejus bafem. Qua
dus fpecies funt feptima & odava.

Sideniq; radices omnes funt impoffibiles vel n,i£
omnes funt reales & inaequales & earum duae funt
affirmative & alterae duae negativae } tunc duae babe-
buntur Hyperbolae ad angulos oppofitos duarum

U u 2 Afymp-

■ . [ 148 ]

Afymptoton cum Hyperbola mguinea circa Afymp-
toton tertiam. Quae fpecies eft nona.

Et hi lunt omnes radicum cafus poffibiles. Nam.
ft duae radices funt aequales inter fe, & alias duae font
etiany inter le aequales, Figura evadet Sedio Conica
cum linea reda.

XVI. Si Hyperbola redundans habet unicam tan turn
d!c^tiuJdZ Diametrum fit ejus Diameter Abfciffa AB, & squa-
ies cum unica tan- tioms hujus ax 3 A~bxx-j-cx J |~d = o quaere tres ra-

turn Dtametro. Ji ces f eu lores X.

17• Si radices illae font omnes reales & ejufdem figni^

Figura conftabit ex Ovali intra triangulum Dd^ ja-
cente & tribus Hyperbolis ad angulos ejus, nempe
circumfcripta ad angulum D & infcriptis duabus ad
angulos d Et haec eft fpecies decima.

184 Si radices duae majores font aequales & tertia ejuf¬

dem figni, crura Hyperbola jacentis verfus D fefe
decuffabunt in forma Modi propter contadum Ova-
lis. Quae fpecies eft undecima.

l 9» Si tres radices funt aequales. Hyperbola ifta fit

cujpdata fine Ovali. Quae fpecies eft duodecima.

2a Si radices duae minores funt aequales & tertia ejuf¬

dem figni, Ovalis in funclum evanuit. Quae fpecies
eft decima tertia. In fpeciebus quatuor noviffimis
Hyperbola quae jacet verfus D Afymptotos in finu
fuo arnpleditur, reliquae duae in finu Afymptoton
jacent.

pf* 20 * Si duae ex radicibus funt impoffibiles habebuntur tres

&£ 22. Hyperbolae fur<e fine Ovali decuffatione vel cufpide.
^23- Et hujus caius fpecies funt quatuor, nempe decima

quarta fi Hyperbola circumfcripta jacet verfus D &

decima

[ H9 J - .

decima quinta fi Hyperbola infcripta jacet verfus D,
decima fexta ii Hyperbola circumicripta jacet fub
bafi cH trianguli DcH, & decima feptima fi Hyper-
bola.infcripta jacet fub eadem bafi.

Si dux radices funt sequales & tertia figni diverfi 2 4*
figura erit cruciformk. Nempeduse ex tribus Hy-
perbolis feinvicem decuffabunt idq; vel ad verticem
trianguli ab Afymptotis comprehenfi vel ad ejus ba-
fem. Qj ix duse lpecies funt decima octave & decima.

nona.

. XVII,
Hyperbola dim

cum

Si duse radices funt inaquales & ejufdem figni &
tertia eft figni diverfi, duse habebuntur Hyperbola
in oppofitis angulis duarum afymptoton cum Con*
choidali intermedia. Conchoidalis autem vel jace- Fi Z* 2 ~
bit ad eafdem partes afymptoti fuse cum triangulo Bg ° 26
ab afymptotis conftituto, vel ad partes contrarias ;

& hi duo cafus conftituunt fpeciem vigefimam & vi¬
gefimam primam.

Hyperbola redundans quae fiabet tres diametros

conftat ex tribus Hyperbolis in finubus afymptot6n re ^Sej.

jacentibus, idq; vel ad angulos trianguli ab afympto- tribus Diametris *
tis comprehenfi vel ad ejus latera. Cafus prior dat p|]
ipeciem vigefimam fecundanq& pofterior Ipeciem vi¬
gefimam tertiam.

Si tres afymptoti in pundto communi fe mutuo XViii.
decuffant, vertuntur fpecies quinta & fexta in vise- H y^oU m -
iimam quartam, leptima & ottava m vigefimam cum Afymptotis
quintam, & nona in vigefimam fextam ubi Anguinea tribu! ad ctmmu ~
non tranfit per concurfum afymptoton, & in vigefi- ZJfmitZ. ° >n ~
rnam leptimam ubi tranfit per concurfum ilium, quo Fi s- 3 P-
cai’u termini b ac d delunt, & concurfus afyrnpto- 32.
ton ell centrum figura ab omnibus ejus partibus 33*

oppofitis

F/>. 34,
Fig. 35.

'%* 37 *

%* .38.

C 1 ^] ,

oppofitis aequaliter diftans. Et hx quatuor fpecles
Diametnim non habent.

Fertuntur etiam fpecies decima quarta ac decima
fexta in vigefimam oftavam, decima quinta ac de¬
cima feptima in vigefimam nonam, decima octava
& decima nona in tricefimam, & vigefima cum vige-
lima prima in tricefimam primam. Et hx fpecies
unicam habent diametrum.

Ac deniq; fpecies vigefima fecunda & vigefima
tertia vertuntur in fpeciem tricefimam fecundam cu-
jus tres funt Diametri per concurfum afymptotdn
tranfeuntes. Quae omnes! converfiones facillime in-
telliguntur faciendo ut triangulum ab afymptotis
comprehenfum diminuatur donee in pun&um eva-
nefcat.

Si in primo aequationum cafu terminus ax 3 ne-
gativus eft, Figura erit Hyberbola defediva unicam
trum nm habrn- liabens afymptoton & duo tantum crura Hyperbo-

lica juxta afymptoton illam in plagas contrarias in¬
finite progredientia. Et afymptotos ilia eft Ordi-
nata prima & principalis AG. Si terminus ey non
deeft figura nullam habebit Diametrum, fi deeft ha-
bebit unicam. In priori cafu fpecies fic enume-
rantur.

Si sequationis hujus ax 4 = bx 5 -j-cxx -[■ dx-ee,
radices omnes At, AP, A A .w, funt reales & in-
oequales, Figura erit Hyperbola anguinea afympto¬
ton flexu contrario amplexa, cum Ovali conjugata.
Quse fpecies eft tricefima tertia.

Si radices duae mediae AP & Af aequentur inter
fe, Ovalis & Anguinea junguntur feie decuflantes
in forma Nodi, Qua’ eft fpecies tricefima quarta.

xix.

tes„

■ £' 3 9 *

• : a. 40.,

[ ]

Si tres radices funt sequaks, Nodus vertetur
cuff idem acutiffimum in vertice anguinese. Et hxc
eft fpecies tricesima quinta.

Si e tribus radicibus ejufdem figni dux maxknae fi £' 43-
A / & A -or fibi rnutuo sequantur, Ovalis in fundtum
evanuit, Q.use fpecies eft tricefima fexta.

Si radices dux quaevis imaginary funt 7 fola ma-
nebit Anguinea fur a, fine Ovalq decuflatione 5 oil-
pide vel pundo conjugate. Si Anguinea ilia non 42*.,

tranfit per pundum A fpecies eft tricefima feptima >
fin tranfit per pundum illud A ( id quod contin^it Fl £* 43*
ubi termini b ac d defunt,) pundum illud A erit
centrum figure redas omnes per ipfum dudas &
ad Curvam utrinq; terminatas bifecans. Et haec
eft fpecies tricefima odava.

In altero cafu ubi terminus ey deeft 8 c propterea xx.
figura Diametrum habet, fi aequationis hums ax 3 H yppp^fip-

i j i J J * ■ a 't* \ a ' r> iZYft dsfeEiiV&.di-

= bXX- r CX+.d radices OmneS AT, At, At, funt amnrum habex,

reales, inasquales & ejufdem figni, figura erit Hyper- p-
bola Conchoidalis cum Ovali ad convexitatem, ""Qua* 4 ’* '
eft fpecies tricefima nona.

Si duse radices funt insequales & ejufdem figni & Bg, 44 ,
tertia eft figni contrarii, Ovalis jacebit ad concavi-
tatem Conchoidalis. Eftq; fpecies quadragefima.

Si radices duse min ores AT, At, funt requales i%. 4 <r,

& tertia At eft ejufdem figni, Ovalis & Conchoi¬
dalis jungentur fefe decuffando in modum Modi,

Quce fpecies eft quadragefima prima.

Si tres radices funt aequales, Nodus mutabitur in ^.47
Cuff idem & figura erit Cijois Veterwm, Et lisec eft
fpecies quadragefima fecunda.

49.

49 *

48,49.

XXL

Hyperbola fep-
tern Parabolica
.Diametrum non
babefft.es.

^ a.

$ig. 51 .

Si radices duse majores funt acquales, & tertia eft
ejufdem figni,Conchoidalis habebit punclum conju-
gatum ad convexitatem fuam, eftq; fpecies quadra-
geiima tertia.

Si radices duae funt sequales & tertia eft figni con-
trarii Conchoidalis habebit punclum conjugatum
ad concavitatem fuam, eftq; fpecies quadragefiraa
quarta.

Si radices duae funt impoflibiles habebitur Con-
choidalis pur a fine Ovali , Nodo, Culpide vel
pun do conjugate. Quae fpecies eft quadragefima
quinta..

Siquando in primo aequationum cafu terminus ax?
deeft & terminus bxx non deeft, Figura erit Hy¬
perbola Parabolica duo habens crura Hyperbolica ad
unam Alymptoton S A G & duo Parabolica in pla-
gam unam 6c eandem convergentia. Si terminus
e y non deeft figura nullam habebit diametrum, fin
deeft habebit unicam. In priori cafu fpecies funt
hae

Si tres radices AP, A-nr, Ax sequationis hujus
bx 3 4 cx +dx-|--' ee = o funt inaequales & ejufdem
figni, figura conftabit ex Ovali & aliis duabus Curvis
quae partim Hyperbolicae funt & partim Parabolics’.
Afempe crura Parabolica continuo duftu junguntur
cruribus Hyperbolicis fibi proximis. Et haec eft
fpecies quadregeiima l'exta.

Si radices duae minores funt sequales & tertia eft
ejufdem figni, Ovalis & una Curvarum illarum
Hyperbolo-Parabolicarum junguntur & fe decuflfant
in formam Nodi. Quae fpecies eft quadragefima
feptima.

C 1 53 ]

Si tres radices funt gequales, Nodus Hie in Cuf- Fig,
pidem vertitur. Eftq; fpecies quadragefima oCtava.

Si radices duse majores funt aequales & tertia eft 53*
ejufdem figni, Ovalis in punttum conjugatum eva-
unit. Qux fpecies eft quadragefima nona.

Si dnx radices funt impoffibiles, manebunt fur<e fi z*% 3,54*
ilia duae curvae Hyperbolo-parabolicae fine Ovali,
decuftatione, cufpide vel pun&o conjugate^ & fpe-
eiem quinquagefimam conftituent.

Si radices duae funt aequales & tertia eft ftgni con - B *' ^

Irani, Curvae illoe hyperbolo-parabolicae junguntur
fefe decuffando in morem crucis. Eftq; fpecies quin-
quagefima prima.

Si radices duae funt incequaies & ejufdem figfii & F ^-
tertia eft figni contrarii, figura evadet Hyperbola
anguinea circa Afymptoton A G, cum Parabola con-
jugata. Et hsec eft fpecies quinquagefima fecunda.

In altero cafu ubi terminus ey deeft & figura xxil.
Diametrum habet, fi dute radices aequationis hujus ttt Z yf F a ™b&7
bxx-j-cx-|-d = o funt impoffibiles, duae habentur P^metrum ha-
figurse hyperbolo-parabolicae a Diametro A B hinc b ZX
inde aequaliter diftantes. Qute fpecies eft quinqua- >7 '
gefima tertia.

Si aquationis illius radices duae funt impoflibiles, Fi £- 58.
figure hyperbolo'parabolicae junguntur fefe de-
cuffantes in morem crucis, & fpeciem quinquagefi¬
mam quartam conftituunt.

Si radices illae funt inaequales & ejufdem figni, ha- Fi £- is-
betur Hyperbola Conchoidalis cum Parabola ex
eodem latere Afymptoti. Eftq; fpecies quinquage¬
fima quinta.

C 1 54 D

*%•*<>: Si radices ilk funt figni contrarii r habetur Con-

choidalis cum Parabola ad alteras partes AfymptotL
Ctuse fpecies eft quinquagefima fexta.
xxiii. f ^Siquando in primo ^quationum cafu terminus

utei ;qj ? x? & b . xx dee . ft v fi g ura erit Hyperbolifmus
b&u, ’ ‘ fedionis alicujus Conicce, Hyperbolifmum figure
voco cujus Ordinata proditapplicandoconteptumfub
Ordinata figure illius & red# data ad AbfciiTam com-
munem. Hac ratione linea reda vertitur in hyper-
bolam Gonicam, & fedio omnis Gonica vertitur in
aliquam figurarum quas hie Hyperbolifmos fedio-
num Conicarum voco. Nam sequatio ad figuras
de quibus agimus, nempe x y y -j- e y = c x -j- d, feu
e+z^eei- ^dx-ft 4 exx generatur appli-

y 2 x

can do contentum fub Ordinata fedionis Conicae

e+/ / ee-b4dx-|-4cxx & reda data m ad curvarum
2 m ' “

AbfciiTam communem x. Unde liquet quod figura
genita Hyperbolifmus erit Hyperbola?, Ellipfeos vel
Parabolae perinde ut terminus cx affirmativus eft
vel negativus vel nullus.

Hyperbolifmus Hyperbola? tres habet afymptotos
quarum una eft Ordinata prima & principalis A d
alterae duse funt parallels Ablciffae A B & ab eadem
bine inde sequaliter diftant. In Ordinata principali
Ad cape Ad, hinc inde aequales quantitati //c
& per punUa d ac ^ age dg, Afymptotos Ab-
feifife A B parallelas.

Ubi terminus ey non deeft figura nullam ha¬
bet diametrum. In hoc cafu fi aequationis hujus
cxx-ydx-j-Jee^o radices duae A P, Ap funt reales

[ !$$ ]

Sc ina'quales (nam tequales efle nequeunt nifi figura
fit Conica feftio ) figura conftabit ex tribus Hyper-
bolis fibi oppofitis quarum una jacet inter afymp-
totos parallelas & altera duae jacent extra. Et haec
eft fpecies quinquagefima feptima.

Si radices illae duae i’unt impoffibiles,habentur Hy¬
perbolae duae oppofitae extra afymptotos parallelas &
Anguinea hyperbolica intra eafdem. Haec figura
duarum eft lpecierum. Nam centrum non habet
ubi terminus d non deeft ; fed fi terminus ille deeft
pun£tum A eft ejus centrum. Prior fpecies eft quin¬
quagefima oftava, pofterior quinquagefima nona.

T Quod fi terminus e y deeft, figura conftabit ex Kg. 64
tribus hyperbolis oppofitis quarum una jacet inter
afymptotos parallelas & altera duae jacent extra ut
in fpecie quinquagefima quarta, & praterea diame-
trum habet quae eft abfciffa A B. Et haec eft fpecies
fexagefima.

Hyperbolifmus Ellipfeos per hanc aequationem de- XXIV.
finitur xy yey = cx-j-d, & unicam habet afymp- Tr ‘t
toton qua eft Ordinata principalis Ad. Si terminus k £ 6 ^'
ey non deeft, figura eft Hyperbola anguinea fine dia~
metro atq; etiam fine centre fi terminus d non deeft.

Quae fpeeies eft fexagefima prima.

^ ^ At ft terminus d deeft, figura habet centrum fine%* 66 *
diametro & centrum ejus eft pun&um A, Species
vero eft fexagefima fecunda,

Et fi terminus ey deeft & terminus d non deeft, Fi g* $7*
figura eft Conchoidalis ad afymptoton A G, habetq;
diametrum fine centro, & diameter ejus eft Abfciffa
A EL Quae fpecies eft fexagefima tertia,

X x 2 Hyper-

xxv.

Duo Hyperbo-
lifmi Parabola.

Fig. 68.
Fig . 6p,

< 6 -

XXVI.

Tridens.

B>e ?6.

XXVII*

Parabola quin
que divergentes.

Fig. 70,71,

73*

[I 5 <S]

Hyperbolifmus Parabola per hanc aequationem
definitur x y y e y = d ; & duas habet afymptotos,
Abfciffam A B" & Ordinatam primam & principalem
AG* Hyperbolae vero in hac figura funtduae, non
in afymptoton angulis oppcfitis fed in angulis qui
funt deinceps jacentes, idq; ad utrumq; latus ab-
fciffse A B, & vel fine diametro fi terminus ey ha-
betur, vel cum diametro fi terminus ille deeft. Qua;
duse fpecies funt fexagefima quarta & fexagefima
quinta.

In fecundo aequationum cafu habebatur sequatio
xy = ax 5 -^bxx-bcx-pd. Et figura in hoc cafu
habet quatuor crura infinita quorum duo jhnt hy-
perbolica circa afymptoton A G in contrarias partes
tendentia & duo Parabolica convergentia & cum
prioribus fpeciem Tridentis fere efformantia. Eftq;
hxc Figura Parabola ilia per quam Cartefius sequa-
tiones lex dimenfionum conftruxit* Hsec eft igitur
Ipecies fexagefima fexta.

In tertio cafu aequatio erat yy = ax 3 ~|-bxx-J-cx
= d, & Parabolum defignat cujus crura divergunt
ab invicem & in contrarias partes infinite progre-
diuntur. AbfciiTa AB eft ejus diameter & fpecies ejus
funt quinq; fequentes.

Si aequationis a x 5 -j- b x a -f- c xd — o radices cm-
nes Ati AT, At funt reales & inaequales,figura eft
Parabola divergens campaniformis cum Ovali ad
verticem. Et fpecies eft fexagefima feptima.

Si radices duae funt aequales, Parabola prodit vel
nadata contingendo Ovalem, vel funStata ob Ovalem
infinite parvam. Quse dure fpecies funt fexagefima
oftava & fexagefima nona,

' Si

[i$7]

Si tres radices funt aequales Parabola erit cufp-&g>7$<
data in vertice. Et haec eft Parabola Neiliana quo?
vulgo femicubica dicitur.

Si radices duae funt impoffibiles, habetur Parabola Fig . 73,74,
fura 'campaniformis fpeciem feptuagefimain primam
conftituens.

I11 quarto cafu aequato erat y- ax -pbxx-f-cx xxvni.

& haec aequatio Parabolam illam WaUiftanam Parabola cuhca.

defignat quae crura habet contraria & cubic a du
ci folet. Et ilc fpecies omnino funt feptuaginta
duae.

Si in planum infinitum a pundto lucido illumina- ^ xxix*
turn umbrae ngurarum projiciantur, umbrae lectio- rum ^vmfouv.
num Conicarum femper erunt fedtiones Conicae, eae
Curvarum fecundi generis femper erunt Curvae fe¬
cund! generis, eae curvarum tertii generis femper
erunt Curvae tertii generis, & fic deinceps in infini¬
tum* Et quemadmodum Circulus umbram proji-
ciendo generat fedtiones omnes conicas, tic Parabolae 4

quinq; divergentes umbris fuis generant & exhi-
bent alias omnes fecundi generis curvas , & fic
Curvae qusedam fimpliciores aliorum generum inve-
niri poffunt quae alias omnes eorundem generum
curvas umbris fuis a pundto lucido in planum pro**
jedtis formabunt.

Diximus Curvas fecundi generis a linea redta in xxx;
pundtis tribus fecari poffe, Horum duo nonnun-
quam coincidunt lit cum redta per Ovalem infi¬
nite parvam tranfit vel per concurfum duarum par-
tium Curvae fe mutuo fecantium vel in cufpidem
coeuntium ducitur* Et fiquando redtae omnes in

plagana

XXXI.

'Theoremata de
Cur varum de-
fcriytione orga~
nica.

[158]

plagam cruris alicujus infiniti tendentes Curvam
in unico tantum pundto fecant ( ut fit in ordinatis
Parabolae Cartefianae & Parabolae cubicse, nec non in
redtis Abfciffae Hyperbolifmorum Hyberbolae & Para¬
bolae ySarallelis ) concipiendum eft quod redtae illae
per alia duo Curvae pundta ad infinitam diftan-
tiam lita ( ut ita dicam ) tranfeunt. Hujufmodi
interfedtiones duas coincidentes five ad finitam
fint diftantiam five ad infinitatn, vocabimus pun-
dtum duplex. Curvae autem quae habent pun*
dtum duplex defcribi poflimt per fequentia Theo«
remata.

1. Si anguliduo magnitudine dati PAD, PBD circa
polos pofitione datos A, B rotentur, & eorum crura
A P, B P concurfu fuo P percurrant lineam redtam ;
crura duo reliqua A D, B D concurfu fuo D defcri-
bent fedtionem Conicam per polos A, B tranfeun-
tem : proeterquam ubi lineailla redta tranfit per po-
lorum alterutrum Avel B, vel anguli BAD, ABD
fimul evanefcunt, quibus in cafibus pundtum D de*
fcribet lineam redtam.

Si crura prima A P, B P concurfu fuo P
percurrant fedtionem Conicam per polum alter-
utrum A tranfeuntem, crura duo reliqua AD, BD
concurfu fuo D defcribent Curvam fecundi gene¬
ris per polum alterum B tranfeuntem & pun¬
ctual duplex habentem in polo primo A per quern
fedtio Conica tranfit : praeterquam ubi anguli
BAD, ABD fimul evanefcunt, quo cafu pun¬
dtum

[ r$9 1

£tum D defcribet aliam fefitionem Conicam per po-
lum A tranfeuntem.

g, At fi feftio Conica quam punfitum P percur-
rit tranfeat per neutrum polorum A, B, punftum
D defcribet curvam fecundi vel tertii generis pun-
Sum duplex habentem. Et punStum illud duplex
in concurfu crurum defcribentium, A D, BD in-
venietur ubi anguli B AP, A BP fimul evanefcunt,

Curva autem defcripta fecundi erit generis fi an¬
guli BAD, ABD limul evanefcunt, alias erit ter¬
tii generis & alia duo habebit punCta duplicia in
polis A & B*

Jam feftio Conica determinate' ex datis ejus xxxn.
pundis quinq; & per eadem fie defcribi poteft.

Dentur ejus punda quinq; A, B, C, D, E. Jun- tkp r dat*. quin-

gantur eorum tria qusevis A, B, C & trianguli ABC r ue ( unna -

rotentur anguli duo quivis C A B, CBA circa ver- V

tices fuos A & B, & ubi crurum A C, B C interfedio

C fucceffive applicatur ad punda duo reliqua D, E,

incidat interfedio crurum reliquorum AB & BA

in punda P & Q, Agatur & infinite producatur

reda P Q, & anguli mobiles ita rotentur ut inter-

fedio crurum AB, BA percurrat redam PQ, &

crurum reliquorum interfedio C defcribet propofi-

tam fedionem Conicam per Theorema primum.

xxxin.

Curvae omnes fecundi generis_ pundum duplex cuSuJZ™
habentes determinantur ex datis earum punctis & um duplex ha-
feptem, quorum unum eft pundtum illud duplex,

& tern pimcta.

[ itfo ]

& per eadem pundta fic defcribl poffunti Dentur
Curvae defcribendse pundta quaelibet feptem A, B, C,
D, E, F, G quorum A eft pundtum duplex, Jun-
gantur pundtum A & alia duo quaevis e pundtis puta
B & C ; & trianguli ABC rotetur turn angulus
CAB circa, verticem fuum A, turn angulorum reli-
quorum alteruter ABC circa verticem fuum B. Et
ubi crurum AC ? BC concurfus C lucceffive appli-
catur ad pundta quatuor reliqua D, E, F, G incidat
concurfus crurum reliquorum A B & B A in pundta
quatuor P, Q, R, S. Per pundta ilia quatuor &
quintum A defcribatur fedtio Conica, & anguli pro¬
fit! CAB, CB A ita rotentur ut crurum AB, B A
concurfus percurrat fedtionem illam Conicam , &
concurfus reliquorum crurum AC } B C defcribet
Curvam propofitam per Theorema fecundum.

Si vice pundti C datur pofitione redta B C quae
Curvam defcribendam tangit in B, lineae AD, A P
coincident, & vice anguli D AP habebitur linea redta
circa polum A rotanda.

Si pundtum duplex A infinite diftat debebit Redta
ad plagam pundti illius perpetuo dirigi & motu pa-
rallelo ferri interea dum angulus ABC circa polum
B rotatur.

Defcribi etiam poffunt hx curvae paulo aliter per
Theorema tertium, fed defcriptionem fimpliciorem
pofuilTe fufficit.

Eadem methodo Curvas tertii, quarti & fuperio-
rum generum defcribere licet, non omnes quidem
fed quotquot ratione aliqua commoda per moturn
localem defcribi poffunt* Nam curvam aliquam

fecund i

E ]

■fecundi vel fuperioris generis pundum duplex non
habentem commode defcribere Problema eft inter
difficiliora numerandum.

Curvarum ufus in Geometria eft ut per earum xxxiv.
interfediones Problemata folvantur. Proponatur
sequatio conftruenda dimenfionum novem x^ft-bx 7 fcnpiomm Cm-
-j-cx 6 -i-dx 5 -|-ex 4 4 -fx 3 -j-gxx-|-hx-j-k = o, Ubi wum -

-ft m

b, c, d, )Sc. fignificant quantitates quafvis datas
fignis fuis ft- & -< affefitas. Affumatur sequatio ad
Parabolam cubicam x ! = y, & sequatio prior, fcri-
bendo y pro x ? , evadet y 5 ft-bxy y ft-cyyft-dxxy
-ft e x y ft- m y ft- f x 3 ft-g x x ft- h x -ft k = o, xquatio ad
Curvam aliam fecundi generis. Ubi m vel f deefife
poteft vel pro lubitu affumi. Et per harum Curva¬
rum defcriptiones & interfediones dabuntur radices
aequationis conftruendse. Parabolam cubicam feme!
defcribere fufficit.

Si xquatio conftruenda per defectum duorum ter- A

minorum ultimorum h x & k reducatur ad feptem
dimenfiones, Curva altera delendo m, habebit pun-
fit um duplex in principio abfciffae, & inde facile de-
lcribi poteft ut fupra.

Si sequatio conftruenda per defefitum termino-
rum trium ultimorum gxxft-hxft-k reducatur ad
fex dimenfiones, Curva altera delendo f evadet
fedio Conica.

Et fti per defedum fex ultimorum terminorum
xquatio conftruenda reducatur ad tres dimenfiones,
incidetur in conftrudionem Wallifimam per Para¬
bolam cubicam & lineam redam.

Conftrui etiam poffunt aequationes per Hyperbo-
filmum Parabola cum dlametro. Ut fi conftruenda
fit hsec sequatio dimenfionum novem tormina penul-
tirno carens, a-pcxxfi-dx 3 4 'ex 4 'rfx ,-j-gx 6 -|-hx 7

^kx 8 -l'lx 9 =o j affumatur sequatio ad Hyperbolif-
mum ilium xxy = i, & fcribendo y pro sequatio
conftruenda vertetur in hanc ay 5 J |- c y y -j- d x yy -|- e y
+ f x y -j- m x x y•«-(-gh x-f k xx'-fi 1 x^ — o ,qux cur-
vam fecund! generis defignat cujus defcriptione
Problema folvetun Et quantitatum m ac g alter-
utra hie deeffe poteft, vel pro lubitu affumi.

Per Parabolam cubicam & Curvas tertii generis
conftruuntur etiam sequationes omnes dimenfionum
non plufquam duodecinq & per eandem Parabolam
& curvas quarti generis conftruuntur omnes dimen¬
fionum non plufquam quindecinq Et fic deinceps in
infinitum. Et curvae illse tertii quarti & fuperiorum
generum deferibi femper poffunt inveniendo eorum
pundta per Geometriam planam. lit fi conftruenda
fit sequatiox IS * d-ax IOJ pbx 9 '|-cx 8 J pdx 7 +ex 6 -|-fx ?
ft-gx 4 fi- hx 3 -p ixx -y kx -j- l = o 5 & deferipta

habeatur Parabola Cubica; fit sequatio ad Pa¬
rabolam illarn cubicam x 3 = & fcribendo y

pro x 2 sequatio conftruenda vertetur in hanc
j 4 -j-axy 3 -J-cxxyy -^fxxy 4 ~ixx = o 5 quae eft

rfb

-|^k x

sequatio ad Curvam tertii generis cujus defcriptione
Problema folvetur. Deferibi autem poteft haec Curva
inveniendo ejus punfta per Geometriam plananqprop-
terea quod indeterminata quantitas x non nifi ad
duas dimenfiones afeendit.

(-n/varum. Tab:I.

Cu/nm/rwrv Tab. H.

✓

Cjjrvwrwnv Tab. IE.

)

>•

•/

/ I

( urvnrum Tab-. 14.

TRACTATUS

D E

Quadratura Cur vanun,

[Itf*]

Q Uantitates Mathematicas non ut ex partibus
quam minimis conftantes, fed ut motu conti-
nuo defcriptas hie confidero. Lines deferi-
buntur ac deferibendo generantur non per appofi-
tionem partium fed per motum continuum pundto-
rum, fuperficies per motum linearum, folida per
motum fuperficierum, anguli per rotationem late-
rum, tempora per fluxum continuum, & fic in
teris. Hs Genefes in rerum natura locutn vere ha-
bent & in motu corporum quotidie cernuntur. Et
ad hunemodum Veteres ducendo reddas mobiles in
longitudinem re&arum immobilium genefin docue-
runt redtangulorum,

Confiderando igitur quod quantitates squalibus
temporibus crefcentes & crefcendo genitae, pro velo-
citate majori vel minori qua crefcunt ac generantur,
evadunt majores vel minores; methodum qusrebam

deter-

1x66-] : _ .

determinandi quantitates ex velocitalibus rnotuum
vel incrementorum quibus generantur ; & has mo
tuum vel incrementorum velocitates nominando Flu¬
xiones & quantitates genitas nominando Fluentes , in-
cidi paulatim Amis 1665 & 1666 in MethodumFlu-
xionum qua hie ufus fum in Quadratura Curvarum.

■ Fluxiones funt quam proxime ut Fluentium aug-
menta sequalibus temporis particulis quam minimis
gen'ita, & ut accurate loquar, funt in prima ratione
dugmentorum nafeentmm; exponi autem poffunt per
lineas quafeunq; quae funt ipfis proportion?;les. Ut
Fig, t fi areae ABC., ABDG Ordinatis BC . B D fuper

bah A B uniform! cum motu progredientibus deferi-
bantur, harum arearum fli xiones erunt inter fe ut
Ordinates defenbentes BC A BD 5 & per Ordinatas-
alias exponi poffunt, propterea quod Ordinatae ilia?
funt ut arearum aug rienta nalcentia. Progre-
diatur Ordinata BC de loco fuo BC in locum
quemvis novum b c. Compleatur parallelogram-
mum BC E b, ac ducatur reda VTH quse Cur-
vam tangat in C ipfifq; be & BA produdis occur-
/rat in T & V : & Abfciffs A Ordinatae B C 5 &
Lineoe Curvae A C c augmenta modo genita erunt
Bb^ Ec & Cc; & in horurii augmentortim nafcen-
tium ratione prima funt latera trianguli CE fpdeoq;
fluxiones ipfarum AB 5 BC & AC funt ut trianguli
illius CET latera CE 5 ET &CT & per eadem
latera exponi poffunt, vel quod perinde eft per la¬
tera trianguli coiiftmilis V B C.

Eodem recidit fi fuinantur fluxiones in ultima
'.ratione par tium evanefeentium. Agatuf reda Cc
& _producat.ur eadem ad K, Redeat Ordinata be

C 167 ]

in locum fuum priorem B C, & coeuntibus ptm£tis
C & c, reda C K coincidet cum tangente C H, &
triangulum evanefcens CEc in ultima fua forma
evadet fimile triangulo GET., & ejus latera evanef-
centia CE, Ec & Cc erunt ultimo inter feut funt
trianguli alterius GET latera CE, ET&CT, &
propterea in hac ratione funt fluxiones linearum A B ?

BC & AC. Si punda C & c parvo quovis inter-
vallo ab invicem diftant reda C K parvo interval!© a
tangente CHdiftabit. lit reda CK cum tangente
C H coincidat Sc ratiories ultimo linearum C E, E c &

Cc inveniantuiq debent punda C Sc c coire & om-
nino coincidere. Errores quam minimi in rebus-,
mathematicis non funt contemnendL

Simili argumento fi circulus centre B radio BC
deferiptus in longitudinem Abfciffae AB ad angulos
rectos uniform! cum motu ducatur, fluxio folidi ge~
niti ABC erit ut circulus illegenerans ? & fluxio fu-
perficiei ejus erit ut perimeter Circuli illius &
fluxio linege curvse A C conjundtim. Nam quo tem¬
pore folidum ABC generator ducendo circulum
ilium in longitudinem AbfcifSe A B, eodem fuper-
ficies ejus generator ducendo perimetrum circuli il¬
lius in longitudinem Curvse A C s
Rjeda TB circa plum datum T revolvensjecet aliam Fig , 2 0
pfitione datum re dam AB: quaritur froprtio fluxio-
mm red arum illarum AB Id T B . Progrediatiir
reda P B de loco fuo P B in locum novum P b* In
P b capiatur P C ipfi P B sequalis 1 & ad AB ducatur
PD fic, ut angulus bPD sequalis fit angulo bBC ;

& ob fimilitudinem triangulorum bBC, bPD erit
augmentum Bb ad augmentum Cb ut Pb ad Db,

Redeat

1*58 3

Redeat jaili P b in locum fuum prioreni P B ut aug-
menta ilia evanefcant, & evanefcentium ratio ulti¬
ma, id eft ratio ultima Pb ad Db, ea erit quae eft
PB ad DB, exiftente angulo PDB redo, & prop-
tereain hac ratione eft fluxio ipfius A B ad fluxionem
ipfius P B.

Recta PB circa datum Polum P revolvens fecet
■ alias duos fo/itione datas rectos y 4 B ly yl E in B
E : qudcntur frofortio fiuxionum reel arum ittarum
ElB . 1 ? AE. Progrediatur reda revolvens PB de
loco fuo P B in locum novum P b redas A B, A E in
pundis b & e fecantem, & redse A E parallela B G
ducatur ipfi Pb occurrens in C, & erit Bb ad BG ut
A b ad A e, & B C ad E e ut P B ad P E, & conjundis
rationibus Bb ad Ee ut AbxPB ad AexPE.
Redeat jam linea P b in locum fuum priorem P B, &
augmentum evanefeens Bb erit ad augmentum eva-
nefeens Ee ut ABkPB ad AExPE, ideoq; in
hac ratione ell fluxio reda: AB ad fluxionem recta'

A E._

Hinc fi recta revolvens P B lineas quafvis Curvas
pofitione datas fecet in pundis B & E, & recta’ jam
mobiles AB,AE Curvas illas tangant in Sedionum
pundis B & E: erit fluxio Curva quam reda, A B
'tangit ad fluxionem Curvae quam reda A E tangit
ut A BxP B ad A E/P E. Id quod etiam eveniet
fi reda P B Curvam aliquam pofitione datam perpe-
tuo tangat in pundo mobili P.

Fluat quantitas x unifermiter l? invenienda fit fluxio
qumtitatis x n . Quo tempore quantitas x fluendo

evadit xj-o, quantitas x n evadet x-j-o| n , id eft
per anethodum ferierum infinitarum, x n -j nox lw

[Itfp]

'-j-«|?oox n ‘ 2 -|-if5V. Et augmenta o & nox a 'H "±£oox®' 2
-\-lfjc. funt ad invicem ut i & nx n ' I -|--2£fox n ‘ 2 -j- y c .
Evanefcant jam augmenta ilia, & eorum ratio
ultima erit i ad nx 11 ' 1 : ideoq; fluxio quantitatis
x eft ad fluxionem quantitatis x n ut i adnx”' 1 .

Similibus argumentis per methodum rationum
primarum & ultimarum colligi poffunt fluxiones li-
nearum feu redtarum feu curvarum in cafibus qui-
bufcunque, ut & fluxiones fuperficierum, angulo-
rum & aliarum quantitatum. In finitis autem quan-
titatibus Analyfin lie inftituere, & finitarum nalcen-
tium vel evanefeentium rationes primas vel ultimas
inveftigare, confonum eft Geometric Veterum: &
volui oftendere quod in Methodo Fluxionum non
opus fit figuras infinite parvas in Geometriam intro-
ducere. Peragi tamen poteft Analyfis in figuris qui-
bufeunq; feu finitis feu infinite parvis qua figuris
evanefeentibus finguntur fimiles, ut & in figuris qua'
pro infinite parvis haberi folent, modo caute pro-
cedas.

Ex Fluxionibus invenire Fluentes Problema dif-
ficilius eft, & folutionis primus gradus aequipollet
Quadrature Curvarum : de qua fequentia olim
feripfi.

D E

TRACTATUS

D E

Quadratura Curvarum.

Q Uantitates indeterminatas ut motu perpetuo
. creicentes vel decrefcentes, id eft ut fluen-
tes vel defluentes in fequentibus confidero,deftgnoq;
literis z 5 y, x ? v ? & earum fiuxiones feu celeritates

9 * * «

erefcendi noto iifdem literis pun&atis z, y, v.
Sunt & harum fluxionum fiuxiones feu mutationes
magis aut minus celeres quas. ipfarum z, y, x 5 v
fiuxiones fecundas nominare licet & fic dignare

*-• I? •• <9%x

z 5 y, x ? v, & harum fiuxiones primas feu ipfarum

9 • * *

» ft «• » . ft * ♦

z 5 y 7 x, v fiuxiones tertias fic z, y 5 x 5 v, & quartas fic

*••••••• •

o,® !5 <1 ft SO

z 5 y, x } v. Et quemadmodum z 5 y, x, v funt flu-

xiones quantitatum z 5 y r v 5 & hoe funt fiuxiones

quantitatum z, y 5 x, v & hoe funt fiuxiones quantita¬
tum primarum z, y, x 5 v : fic hoe quantitates confide-
rari poffunt ut fiuxiones aliarum quas fic defignabo,

[ I 7 I ]

z, y, x, v, & hse utfluxiones aliarum z,y 5 x, v, &
hoc ut nuxiones aliarum z*> y 5 x., v. Dengnant lgitur

j «• «®

f * • «• * # •* ®«i;

z 5 z, z, z, z, z } z 7 z r tor. feriem quantitatum quarum
quoelibet pofterior eft fluxio prsecedentis & quaelibet
prior eft fluens quantitas fluxionem habens fubfe-

r 1 1 f . J- .-r-X X, CS

quentem. Similis eft feries //az—zz ? //az—zz ?

f ~ 1 <-• 1 - wu . f - ■ - r, -nn' # -w.i.*.-*a=J - - ,-iS 8 @ —■—^»«=a

:az—zz , f/ az—zz , V\ az—zz, ^az—zz, ut &
_ az-|-z 2 az-f-z 2 az-pz 2 az-j-z 2 az-j-z 2

a—z

Si Si if fj

•z

az-f-z 2

. Et notandum eft quod quantitas quaelibet

prior in his feriebus eft ut area figurae curviliniae
cujus ordinatim applicata re&angula eft quantitas

pofterior & abfcifla eft z : uti //az—zz area curvae

cujus ordinata eft / az—zz & abfcifla z. Quo au«*
tem fpedant haec omnia patebit in Propofitionibus
quae fequuntur.

Zz a

PROP,

[ r 7 2 ]

PROP. I. PROB. I.

2 )at a <equatione quotcunq; fluent es qmntit-ates invol- -
vente^ invenire fluxiones.

Solatia .

Multiplicetur omnis sequationis terminus per in-
dicem dignitatis quantitatis cujufq; fluentis quam
involvit, & in fingulis multiplicationibus mutetur
dignitatis latus in fluxionem fuam, & aggrega-
tum faftorum omnium fub propriis lignis erit
equatio nova.

Explicatio .

Sunto a, b, c 9 d l9V\ quantitates determinate &
immutabiles, 8c proponatur equatio quevis quan¬
titates fluentes z, y 5 x 1 Sc. involvens, uti x 3 — xyy
-j-aaz —b ? == o. Multiplicentur termini primoper
indices dignitatum x, 8c in fingulis multiplication^

bus pro dignitatis latere^ feu x unius dimenfionis,
• • •

fcribatur x ? 8c fumma fadorum erit 3 x x 2 —x y y.Idem

dibit a a z. Ponatur fumma fadtorum equalis ni-

8 » ©

hilo, Sc habebitur equatio 3 x x 2 —xyy — 2 x y y

-]-aaz = o. Dico quodhac cecjuationedelinitur re-
latio fluxionum.

De-

[ i73l

Demonftratio.

Nam fit o quantitas admodum parva & fun to
* * •

oz, oy, ox, quantitatum z, y, x momenta id eft in-
crementa momentanea fynchrona. Et fi quantita¬
tes ftuentes jam funt z, y & x, hae poft momentum

» x> ®

temporis incrementis fuis oz, oy, ox aufite, evadent
• « •

z-|-oz, y-^oy, xq-ox, quae in aequatione prima pro

z, y & x fcriptae dant aequationem x 3 -j-^xxox
• • • *

3X00XX o 3 x3 —xyy —oxyy—2xoyy — ixooyy

a 9 • ® * •

—xooyy—xo 3 yy-j-aaz^-aaoz—b^ = o. Subducatur
aequatio prior, & refiduum divifum per o erit 3XX2

9 V » 9 '"' 9 * ® ® e • A

•^3xxox~^x 3 oo — xyy— 3 xyy —^xoyy — xoyy —xooyy

4 -aaz = b. Minuatur quantitas o in infinitum,& neg~

* »

leftis terminis evanefcentibus reftabit gxx 2 —xyy
» •

—2 x yy T aaz —°- Q- e. d,

_ «

Explicatio plenior.

Ad eundem modum fi asquatio effet X 3 —xyv

^E2S3ca=s=ssasK=s=3a=9 ^ v • %

-J-aa f^ax—yy—b^ — o, produceretur :jx 2 x-—xyy -
• * ' • .

~ 2 xyy+aa^ax—yy = o. Xlbi fi fiuxionern i/\ a?.—yy

tollere velis, pone Kax—yy = z, & erit ax—yy = z a

[i 74 ]

# ♦ ' ♦

& (.per hanc Fropofitionem ) ax— !2 yy = ? /:i feu

ax -^yy

= z, hoc eft

ax—ayy

^ax—-jry • E

ax—yy

. * • ,, a 3 x—iaayy

inde g x 2 x—-xyy— 2 xyy -p — . ~=o

2 / ax—-yy

Et per operationem repetitam pergitur ad fiuxio-
mes fecundas, tertias & fequentes. Sit sequatio
zy 3 —Z4-|-a4 = o, & fiet per operationem primam

-* • t « * *

-zy^Vim 2 — A- zz ?"°-> per fecundani zy 3 -j-6zyy 2
®« • «« *

+3 z yy 2 + 6z y 2 y—4 ZZ3 “ I ° , per tertiam

- c®# 9 *S •» *** **"*>

zy 1 v 9zyy 2 + 9*yy*-h i8 w-V3 z yy 2 -l- l8z yyy

8 9 9 *19 ©

6zy 3 —qzz3—3 6zzz 2 — aqz 3 z=o.

tlbi vero fic pergitur ad fluxiones fecundas, ter¬
tias & feqtientes, convenit quantitatem aliquam ut
uniformiter fiuentem conliderar.e T & pro ejus iluxione
prima unitatem teribere, pro fecunda vero & fe-
quentibus nihil. Sit aequatio zy 3 — z 4 -p a4 = o, ut
iupra ; Sc fluat Z uniformiter, fitq; ejus fluxio unitas,

8 c fiet, per operationem, primam y^-^syy 2 —4Z3 = o,

■ * «« «

; .per fecundam 6yy 2 -ji 3zyy 2 ~^6zy 2 y — iaz 2 = o,

.per tertiam 9 yf+ i Bpy+JzyyH’ iS^yy-j-^zyg
-•*-—aqz = o.

i i 7$ ]

In hujus autem generis aequationibus concipien-

dum eft quod ftuxiones in fingulis terminis fint ejul-

® *

deni ordinis, id eft vel -omnes primi ordinis y, z,

• • » • • 9

vel omnes fecundi y, y 2 9 yz ? z 2 , vel omnes tertii
••• »• • • • « •• •

y, yy, yz, y 3 , y 2 z, yz 2 Z 3 &c. Et ubi res aliter fe
habet complendus eft ordo per fubintelle&as ftuxio¬
nes quantitatis uniformiter fluentis, Sic aequatio

« « »

noviffima complendo ordinem tertium fit 2
J f 18zy 2 y-|-3Zyy 2 -' r 1 8 zyyy-j-, 6 zy 3 —aqzz? = o.

PROP. II. PROB.IL

Invmire Curvets qua qmdrari poffunt.

Sit ABC figura invenienda, BC Ordinatim ap- Erg. 4. '

plicata reftangula, & AB abfeifla, Producatur
CB ad E ut fit BE— 1 ? & compleatur parallelo-
grammum ABED: & arearum ABC, ABED
ftuxiones erunt ut BC &BE, Aftumatur igitur
aequatio quaevis qua relatio arearum definiatur r &
inde dabitur relatio ordinatarum BC &BE per
Prop. I. Q E, L

Hujus rei . exempla habentur in Propofitionibus
duabus fequentibus.

PR OR

C176]

PROP. III. THEOR. I.

Si pro abfcifla A B ,8i area AE feu ABxi pro-
milcuefcribaturz 5 & fipro e -Egz 2fl -|-hz 3 «-j-’&c s
fcribatur R: fit autem area Curvae z®R h erit.
ordinatim applicata BC —

&c. in z 6-1 R K ~ l .

Demonjlratio.

Nam fi fit z8R ,i =V, erit per Prop. 1 , 9 zz 6 ' ! Ra
® • ®

^-^z 8 RR VI = v. Pro R* in primo aequationis ter-

mino & z 9 in fecundo fcribe RR VI & zz 9 ' 1 , & fiet

« * •

§zR-j-*zR in z 8 '*R\ r = v. Erat autem R=e-|-fz»

~\-gz 2 «-\-hz^ &c. & inde per Prop. 1 . fit R =j

® • ®

^fzz^ 1 J p2„gzz 2M ' I -|-’3whzz 3 «‘ i >'|- &c. quibus fubftitu-

tis & fcripta BE feu 1 pro z 7 fiet

in *M R

Q. E. D.

Vl = v=BC.

PROP.

[ 3

"in

PROP. IV, THEOR, II.

Si Curvae abfciffa A B fit z, 8 c fi pro e-j-> fz» -j-gz 1 *
8 cc. fcribatur R, 8 c pro k-}-lz»-}-mz 2 * 4 - &c. fcri-
batur S ; fit autem area Curvse z® R* : erit or-
dinatim applicata BC=,

| S jgW “^0 ®^S 9 ® 89 ^*»e*

4" 0 0 1 7H

-ta« c 1 z

i f 1 z *" P» S lz3 " *. I i^R^S***

±l„ eraz2 " il ftnz3, ZL § mz40 -*

i-2f« 1' 2 ^S

Demonftratur ad modum Propofitionis fuperioris.

PROP. V. THEOR. III.

SiCurvae abfciffa AB fit z, & pro e-|-fz , '-j-gz I *
-’p hz 3 " -j- 8 cc. Icribatur R : fit autem ordinatim ap¬
plicata z 9 l R VI in a -f-bz" +cz 2 » -[-dz 3 "-|- 8 cc. 8 c po-

natur ? = r. r-f-*=s. s 4 -* = t. t 4 -* = v. 8 cc. erit area

2 *R> i„ ;a A ilMfA .. : • c -. fB -'S A . , >‘ d r.fC^gB-vhA

- T -T T_- Z" T——— ■ - z 2 " + —- ■ Z3»

r —f- 1 9 e

" ' * 5 “ r *4— 2 j e r —l.

—s f r\ —t p —v L n

H- ~ 3 ~ 2g z*” -I- 8 cc. Ubi A, B, C, D, 8 cc.

3 ?e

r-H 4 ,e

Aaa

denotant

i. «eA l|
a ■ * ‘ '<R

C 1 7 8 3

denotant totas coefficientes datas terminorum fingu-
lorum in ferie cum fignis luis -\ •&—,nempe A primi

termini coefficientem jlE B fecundi coefficientem

fcfA C tertii coefficientem

r —j- i, e

r-+ 2 ,e

lie deinceps.

Demonjlratio.

Sunto juxta Propolitionem tertiam 3

Curvarum Ordinatse

fAz" tl gAz 2 " \\ } hAz 3 "&c. V

■'AH "1-3 AH 1

;,eBz"+® 1 '»fBz 2, '1 _s ; ; |~"gBz 3 "&c. i

"r aw n~ 2 a«o r z 9-xT^>r

+g+ 2 »,eCz 2 ''ji H' 2 "fCz 3 " &c.

e c « e

•+8-h»,eDz 3 "&c.j

& earundem ares,
Az 9 R\

Bz 9 +” R .

Cz 9 + 2 » R A .
Dz 9 '^ Ra.

Et (i fitmma ordinatarum ponatur aequalis ordi¬
nate a^kzM'C^d'dz 3 "-^ &c. in z 9 ' 1 R A '\ fumma
a rearum z 9 R A in A ^Bz» ^Cz 2 «4~Dz 3 » , d- &c. equa-
lis erit area? Curve cujus ifta eft ordinata. iEquen-
tur igitur Ordinatarum termini correfpondentes, &
fiet a=eeA, b = + 9 „fAeB, c= +2 L§^ fB

'4- FF*T,e C &c. & inde $; = A. ^4—— j

• 1 5 _______ ^ 0-h« 3 e

C s=== i§"l ~1 rl A/jji B z’ j * » »

-- — 7 f^ 2h e ~~- iy= Cu Et lie deinceps m infi¬
nitum

[i?9]

nitum. Pone jam £ = r. r-|-A = s. s-j-A = t &c. &

in area z 9 R A x A^-Bz w 'jr-Cz 2 w -j-Dz 3B &c» fcribe ip-
forum A, B, G, &c. valores inventos & prodibit
feries propoiita. Q. E. D.

Et notandmn eft quod Ordinata omnis duobus
modis iu feriem refolvitur. Nam index» vel affir-
mativus eft poteft vel negativus. Proponatur Ordi¬
nata ' —1 - . Hac vel fxc fcribi poteft

z~»x3k—Izzxk —-lzz^-mz^Tq vel lie zx -l-j-^kz'*

xm-lz _ 1 'l-kz _i j -i. In cafu prioreeft a~ gk.b = o.
c== —k e=k. f=o. g=-l. h—m. *=~i. »—i.
s-i=-b 9 =-i=r. s=-i. t=_i. v=o. In
pofteriore eft a=— 1 . b=o. c=3k. e—m. f=—l.
g=o, h=i. A =-j. «=— i. e— i = i. »=2. r=—2.
s=—i;. t==—x. v— Tentandus eft cal’us uter-
que. _ Et fi ferierum alterutra ob terminos tandem
deficientes abrumpitur ac terminatur, habebitur area
Curva? in terminis finitis. Sic in exempli hujus
priore cafu feribendo in ferie valores ipforum a, b,
e, e, f, g, h, a, s y r, s, t, v, termini omnes poll pri-
mum evanefeunt in infinitum & area Curvae prodit

ay k ~~d-™’_ p; t } ia;C area ob lignum negativum
adjacet abfeifise ultra ordinatam produdtse. Nam
area omnis affirmativa adjacet tain abfcilTae quam
ordinata?, negativa vero cadit ad contrarias par¬
tes ordinata & adjacet abfeilfe produtlse, manente
fcilicet figno Ordinatae. Hoc modo. feries alter¬
utra & nonnunquam utraque femper terminatin'
& finita evadit fi Curva geometrice quadrari po¬
teft. At fi Curva talem quadraturam non admit-
tit, feries utraq; continuabitur in infinitum. & ea-

A aa 2 rum

[ i8o ]

rum altera converget & aream dabit approximando,
praeterquain ubi r (propter aream inffoitam) vel
nihil eft vel numerus integer & negativus, vel ubi i-

aequalis eft unitati. Si * minor eft unitate, conver-

get feries in qua index „ affirmativus eft : fin l unita

te major eft, converget feries altera. In uno cafu
area adjacet abfciffe ad ufq; ordinatam duda?, in
altero adjacet abfciffae ultra ordinatam produdae.

Nota infuper quod fi Ordinata contentum eft fub
fadore rationali Q & fadore furdo irreducibili R*,
& fadoris furdi latus R non dividit fadorem ratio-
nalem Q; erit a-i =tt & R*- 1 = R» Sin fadoris fur-
di latus R dividit fadorem rationalem femel, erit
*—1 = ‘jr-y i & R*- 1 : fi dividit bis, erit

1=7& R V1 ==R*'\~ 2 : fi ter, erit i=rr-|p^
& R/- 1 —R^' 3 : 8 c fic deinceps.

Si Ordinata eft fradio rationalis irreducibilis cum
Denominator ex duobus vel pluribus terminis com*-
pofito : refolvendus eft denominator in divifores
fuos omnes primos. Et fi divifor fit aliquis cui
nullus alius eft aequalis, Curva quadrari nequit:
Sin duo vel plures fint divifores aequales, rejicien-
dus eft eorumunus, 8c fi adhuc alii duo vel plures
fint fibi mutuo aequales 8c prioribus inaequales, re-
jiciendus eft etiam eorum unus, 8c fie in aliis omnir-
bus aequalibus fi adhuc plures fint : deinde divifor
qui relinquitur vel contentum fub diviforibus omni¬
bus qui relinquuntur, fi plures funt, ponendum eft
pro R, 8c ejus quadrati reciprocum R' 2 pro R^,prae-
terquam ubi contentum illud eft quadratum vel cu-
bus vel quadrate quadratum,8cc. quo eafu ejus latus

ponem

[i8i]

ponendum eft pro R & poteftatis index 2 vel 5 vel 4
negative fumptus pro a. & Ordinata ad denomimu-
torem R z vel R 3 vel R 4 vel R ? &c. reducenda.

Ut ft ordinata fit —j fel ZJ ~ 8z 3— ; quoniam hxc

Z5-J-Z4—5Z3—z2-|-8z—4 7 1

fraftio irreducibilis eft & denominatoris divifores
funt pares, nempe z—1, z—1, z—1 & z^a,

z-\- 2 , rejicio magnitudinis utriufque diviforem
unum & reliquorum z—1, z—1 , z\-2 conten-
turn z 3 -—’^z-j—2 pono pro R & ejus quadrat! re-

ciprocum qp feu R— 2 pro R*" 1 . Dein Ordina¬
ta m ad denominatorem R 2 feu R 1 -* rediico, & fit
z 6 — 9z 4 '|-8z 3

z 3 —^zipoj quad.

, id eft Z 3 X 8 - 9 z 4 -z 3 x a — 3 z 4- z 3 f 2

1, &c.

Et inde eft a —8. b=— 9. c — o. d
e=a. f=- 3 , g=o. h=i. k~i = ~7. k = _n
1. 9-1 = 5. s = 4 = r. s= 3. t = a. v=r. Et his
in ferie feriptis prodit area
nibus in tota ferie poft primum evanefeentibus

terminis om-

Si deniq; Ordinata eft fradio irreducibilis & ejus
denominator contentum eft fub fadore rationali Q.
& fadore furdo irreducibili R’, inveniendi funt la-
teris R divifores omnes primi, & rejiciendus eft di-
vifor unus magnitudinis cujufq; & per divifores
qui reftant, fiqui fint, multiplicandus eft fador
rationalis Q.: & ft fadum asquale eft lateri R vel
later is illius poteftati alicui cujus indexed: numerus
integer, efto index ille m, & erit *,~i = m, &

R V. = R" T " m » m fi Ordinata fit

<q<r-~XX pc ub. qj-q<qx—.qxx—x 5

quoniam

[ i82 3

quoniam fa&oris furdi latus R feu q^ -|-qqx—qxx—x^
divifores habet q+x } q-\-x, q—xqui duarumfunt
magnitudinunq rejicio diviforem unum magnitudi-
nis utriufq; & per diviforem q-)~x qui relinquitur
multiplico faftorem rationalem qq—xx. Et quo¬
niam fafihim q^-^qqx—qxx—x^ aequale eft la-
teri R^pono m=i. & inde, cum tt fit fit a— i =

Ordinatam igitur reduco ad denominatorem R:i

& fit Z° x 3qM- aq 5 xi 8q 4 xx l-8q 3 x 3 ~7qqx 4 ~,6qx f
xq^pqqx—qxx—x 3 h*.Unde efta- 3q 6 .b = aq*&c B
e~q3. f — qq &c. 0—1=0. e=i= M . A = —]• r=i.
s = 5 . t- 1 . v = o. Et his in ferie fcriptis prodit

3 3

3 qqx-|- 3 X 3

area —__m. terminis omnibus in ferie tota

, v /cub. a 3 -|-aax—axx—-x3 >

poft tertium evanefcentibus.

PROP. VI. THEO.R. IV.

Si Curvse abfciffa A B fit z, & fcribantur R pro
e J[-fz» Jpgz^^hz 311 --]- &c. & S pro k -j- lz» \-njz 2 »
-\-nz 3B &c. fit autem ordinatim applicata z^'R^ 1 S'- 1 ' 1
in a-Wbz» -\~cz 2 » -\-dz 3 » &c. & fi terminorum, e, f,
g, h, &c. & k, 1, m, n. &c. reftangula fint.

,ek

g l

&c.

[*%]

Et fi re&anguloriim illorum coefficients nuine-
rales lint refpeftive

»9 = r. = s. = t. t -]-a = v. &c,

r--^ = s. s-f> = t. = v. v-| - ^ — w.&c.

S-|- |K = t. — v. v-\-^ = w. W-f-(*=X.&C.

y/ y// JW JUU m a Hi

t~j-#*=v. V~rj-/* = wv = x. = y. &c.

area Curvoe erit hxc

*“~t g k

- o AU—sfkA l n —-s -l-i, fk 73 *—t' f 1 \

ar, a ■ » a » b — s'el A » C -s'+i,el ^-t"em A

z'fiObf 4 in •—_j- —-——— --—

rek * r-j-i, e k

__ —V h k A

_ *~ Z -J-!, g kj^ _ V ' g 1

“-s-|- 2 , fk(^ *—£-|-i, f 1 -D — v "fm
n Al — s %j- 2 , e 1 v<# C'-j-ije m —v'"e n

i-|- 2 , e k

H~3> <? k

Z 3 " &.C

Ubi A denotat termini primi coeffieientem datapn

sL cum figno fuo -\-> vel —, B coeffieientem datam

fecundi, C coeffieientem datam tertii, & fie deinceps.
Terminorum vero, a, b, c, &c. k, 1 , m, &c. unus
vel plures deeffe poffunt. Demonftratur Propofitio
ad modum prsecedentis, & quae ibi notantur hie ob-
tinent. Pergit autem feries talium Propofitionum in
infinitum, & Progreffio feriei manifefta eft.

PROP.

C184]

PROP. VII. THEOR. V.

Si pro e 4 -fz w ~[rgz 2 M-* &c. fcribatur R ut fupra ? &
in Curvae alicujus Ordinata R k A maneant
quantitates datae », A, e 5 £, g, See. & pro <7 ac T fcri-
bantur fucceffive numeri quicunq; integri : Sc ft
detur area unius ex Curvis quae per Ordinatas in-
numeras ftc prodeuntes defignantur ft Ordinatae funt
duorum nominum in vinculo radicis, vel fi dentur
areae duarum ex Curvis fi Ordinatae funt trium no^
minum in vinculo radicis, vel areae trium ex Curvis
fi Ordinatae funt quatuor nominum in vinculo radi-
cis, Sc fic deinceps in infinitum : dico quod dabun-
tur areae curvarum omnium. Pro nominibus hie
habeo terminos omnes in vinculo radicis tarn de-
ficientes quam plenos quorum indices dignitatum
funt in progreffione arithmetica. Sic ordinata

\/a 4 —ax 3 4"X 4 ob terminos duos inter a 4 Sc —ax 3
deficlentes pro quinquinomio haberi debet. At

binomium eft Sc \/a 4 ~H x4 —'It tanoniurm

cumprogreffio jam per majores differentias proce-
da£. Propofitio vero fic demonftratur.

C A S. L

Santo Curvarum duarum Ordinatae pz 8 ' 1 R *- 1 &
qz 6 f rI R A “ E 5 Sc areae pA Sc qB 5 exiftente R quanti¬
tate trium nominum e~j-fz» gz 2 ». Et cum per

Prop.

[■ 85 ]

Prop. III. fit z 9 R* area cmvx cuius Ordinata eft
eel 0 fz" pz 2 * in z^R^.fubduc Ordinatas & areas
priores de area & Ordinata pofteriori, & manebit
9 c gz 2 " in z VI R VI Ordinata nova Curvse, &

—ft Z H

z 9 p*_pA—qB ejufdem area* Pone ee-=p &
=’q & Ordinata evadet gz 2 ® in z 9 R A &
area z s R x —aeA—afB—»»fB. Divide utramq* per
# g-t ,A *g» & aream prodeuntem die C, & affumpta
utcunq; r, erit r C area Curvse cujus Ordinata eft
rjre-Hn-'gvi; Et qua ratione ex areis pA & qB
aream rC Ordinata; rz 9 '^”' 1 R *-' 1 congruentem inve-
nimus, licebit ex areis qB & rC aream quartam
puta sD, ordinatae sz 9 ' 3 "' I R' l ‘ I congruentem invenire,
& fic deinceps in infinitum. Et par eft ratio pro-
greffionis ab areis B & A in partem contrariam
pergentis. Si terminorum 9 , e aliquis de¬

ficit & feriem abrumpit, affumatur area pA in prin-
cipio progreffionis unius & area qB in principio al-
terius, & ex his duabus areis dabuntur ares omnes
in progreffione utraque. Et contra, ex aliis duabus
areis aflumptis fit regreffus per analyfin ad areas A
& B, adeo ut ex duabus datis cseterae omnes den-
tur. Q. E. O. Hie eft cafus Curvarum ubi ipfius z
index o augetur vel diminuitur perpetua additione vel
fubdudtione quantitatis». Cafus alter eft Curva¬
rum ubi index x augetur vel diminuitur unitatibus.

Bbb

CAS,

[i86]

C A S. II.

Ordinate pz 9 ' x R A & qz^^R*, quibus areae pA
& qB jam refpondeant, ft in Rfteu ejj-fz^gz 2 ” du«
cantur ac deinde ad R viciffim applicentur, eva-

dunt pe -j- "h Pg z2?l x z^R^ 1 & qez 11 -J- qfz 2 «
-Uqgz 3 ” x z 9- ^^ 1 . Et per Prop. III. eft az 6 R^
area Curvae cujus Ordinata eft sae ~j_ 9 ^ afz”qt 9 A)j agz 2 «
in z 9 -i R a ' 1 j & bz 8 + M R A area Curvas cujus ordinata
eft ^bez® Il 9 bfz 2rt jl a bgz 3w in z 9l R AI . Et harum qua-

1 —Ail —

tuor arearum fumma eft pA-\-q@rj- az @ R A -|- bz 9 i *R A
& fumma refpondentium ordinatarum

a ae

+pe

+ 0 afz M

-rhn

I" 9 be
ft*

ft pf
+ qe

-|-2AM

jigZ 2 "

-s bf

'+pg

+ qf

9 bgz 3H in z^R*" 1 .

-2A»

+ 02

Si terminus primus tertius & quartus ponantur fe-
orfim sequales nihilo, per primum fiet eae-|-pe=o
feu -8a=p, per quartum - 8b - »b - = q, & per

tertium (eliminando p & q) “/=b. Unde fecundus

fit adeoq; fumma quatuor Ordinatarum eft

—7® e fumma totidem refpondentium

arearum eft az 9 R A -\- 2|f z 8-biR*_8aA— ■ 2 ^~ 2r ^* u> agB.

Divi-

C187 ]

Dividantur hae fummse per > -" aff ~' j P l " a h ^ & }j Quotum
pofterius dicatur D 5 erit D area curvas cujus ordi-
nata eft Quotum prius z®+ 1hI R A ' 1 . Et eadem ratione
ponendo omnes Ordinatae terminos praster primum
asquales nihilo poteft area Gurvae inveniri cujus Or-
dinata eft z 8 l R VI . Dicatur area ifta C, & qua ra¬
tione ex areis A & B invents funt areas C ac D. ex
his areis C ac D inveniri poffunt alias duas E & F
ordinatis z^R*' 2 & z°+ w ' I R v2 congruentes 5 & fic de-
inceps in infinitum. Et per analyfin contrariam
regredi r licet ab areis E&Fad areas CacD, &
inde ad areas A & B, aliafq; quas in progreffione fe-
quuntur. Igitur fi index x perpetua unitatum ad¬
ditione vel lubdudione augeatur vel minuatur, &
ex areis quae Ordinatis fic prodeuntibus refpondent
duas fimpliciffimas habentur ; dantur alias omnes in
infinitum. Q. E. O.

C A $. III.

Et per cafus hofce duos conjunctos^ fi tarn in¬
dex 8 perpetua additione vel fubduftione ipfius n
quam index \ perpetua additione vel fubdufitione
unitatis, utcunq; augeatur vel minuatur, dabuntur
areas fingulis prodeuntibus Ordinatis refpondeiiteso

Q.E.O.

B b b 1

CAS.

C *88 J

CAS . IV.

Et fimili augmento fi ordinata conftat ex qua-
tuor nominibus in vinculo radical! & dantur tres
arearunq vel fi conftat ex quinq; nominibus &
dantur quatuor arearum, & fic deinceps : dabun-
tur areae omnes quae addendo vel fubducendo nume-
rum * indici $ vel unitatem indici x generari poflunt.
Et par eft ratio Curvaruin ubi ordinatae ex binomiis
conflantur, & area una earum quae non funt geome-
trice quadrabiles datur. Q. E. O.

PROP. VIII. THEOR, VL

Si pro e^-fz w 4 'gz 2 "~|~’&c. & k + \z n -j-mz 2 ^-|-&:c,
fcribantur R & S ut fupra 5 & in Curvae alicujus Or¬
dinata z 9 '^ R*± 7 maneant quantitates datae ^
^ ^ /q e, f, g, k, 1, m ? &c. & pro & y) fcri¬
bantur fucceflive numeri quicunq; integri : & fi
dentur areae duarum ex curvis quae per ordinatas
fic prodeuntes defignantur fi quantitates R & S funt
binomia, vel fi dentur areae trium ex curvis fi R
& S conjunftim ex quinq; nominibus conftant, vel
areae quatuor ex curvis fi R & S conjundtim ex fex
nominibus conftant, & fic deinceps in infinitum :
dico quod dabuntur areae curvarum omnium.

Demonftratur ad modum Propofitionis fuperioris.

PROP.

[ iBp]

PROP. IX. THEOR. VII.

JEquantur Curvarum arete inter fe quarum Or-
dinatae font reciproce ut fluxiones Abfciffarum.

Nam contenta fob Ordinatis & fluxionibus Ab¬
fciffarum erunt aequalia, & fluxiones arearum font
ut ha?c contenta.

COROL. I.

Si affumatur relatio quaevis inter Abfciffas dua-
rum Curvarum, & inde per Prop. i. quteratur
relatio fluxionum Abfciffarum, & ponantur Ordi¬
nate reciproce proportionales fluxionibus, inveniri
poffunt innumerte Curvse quarum arete fibi mutua
sequales erunt.

COROL. II.

Sic enim Curva omnis cujus htec eft Ordinata
z e-i i n e f z » g z 2 » &c.|* affumendo quantitatem
quamvis pro » & ponendo j=s & z s = x, migrat in
aliam fibi sequalem cujus ordinata eft £ x f ir» in
e -\- f x" gx 2 " -j-r &c. j A -

[ 19 ° 3

COROL. III.

Et Curva omnis c ujus Ordinata eft z 9 ' 1 in
a bz» ~p cz 2 " -\- & c. x e-p-fz»-p gz 2 " &cj%affumen-
do quantitatem quamvis pro " & ponendo n=s &

V x, migrat in aliam fibi cequalem cujus ordinata
eft in a-p bx'-ft- cx 2 t -p&c.xe-pfx'-pgx 2 '-p&c.f •

COROL. IV.

Et Curva omnis cujus Ordinata eft z®' 1 in
a + bz M cz 2 » -j- &c. x e -|- fz» gz 2 « &c~j*
x k -j- lz M ~]-r mz 2w Scc.J^ affiimendo quantitatem
quamvis pro ? & ponendo 1 = s & z s ~x ? migrat in
aliam fibi sequalem cujus ordinata eft jx-^ina-j-bx,
-j- cx 2v -p & c • x ej f fx" _p gx 2 '-p &c.px k-pix'-J-mx^-p

COROL. V.

Et Curva omnis cujus Ordinata eft z 9 ' 1
pfz" -p gz 2 » 4- &c* ponendo i= x

migrat

aliam fibi aequalem cujus ordinata eft x e -pfx'“

-Pgx 2 » -p Sc? M eft xife * {l duofimt

nomina in vinculo radicis vel x g-p fx*-p ex iB |

fi tria funt nomina ; & fic deinceps.

CO-

[ l 9 l ]

CO ROL. VI,

Et Curva omnis cujus Ordinata eft z u in
e -j- f z" gz 2 ” &c.j* x k lz 8 -\- mz 2 “|- &c.[,“
ponendo i = x migrat in aliarn libi aequalem cu¬
jus ordinata eft —ppj x e -ft fx' fl gx" 28 -|- dcc.j' 1

xM-lx- 8 -^ mx -^-i-&c.l“ id eft Iiy X

xl-j~kx"f* ft bina funt nomina in vinculis radicum,

vel xH-i+aft-bt* x g fx» - j- ex 2 “ft x 1-j-kx"/' 2 ft tria

funt nomina in vinculo radicis prioris ac duo in
vinculo pofterioris: & fie in aliis. Et nota quod
area duae aequales in noviffimis hifce duobus Co-
rollariis jacent ad contrarias partes ordinata rum.
Si area in alterutra curva adjacet abfciffe, area
huic aequalis in altera curva adjacet abfcilfse pra-
ducta.', '

CO ROL. VII.

Si relatio inter Curvse alicujus Ordinatam y &
Abfciffam z definiatur per sequationem quamvis
feftam hujus formas,v“in e-I- fvz^-S- q v^z 20 ' 1 -!-

+ &c. - #ta k ?i r *VV£ + fa y L

figure affumendo 8=^, x=jz^x=^, migrat
ia a ham nbi aequalem cujus Abfciffa x 5 ex data

Ordinata

C192 ]

Ordinata v , determinatur per aequationem non §
affeftam x e-j- fv*q- gv 2M -\- hv 3n &c.i A x k -\- lv#

-p mv 2w -|- &c.f* = x.

COROL . VIII.

Si relatio inter Cur vx alicujus Ordinatarn y &
Abfciffam z definitur per aequationem quamvis
affeftam hujus formas, y a in e^-’fy w z cfS --j“gy 2Mz2cP -f-Scc.

. = z£ in k ly»z d ' 4^my 2 «z 2 ^--&c -$~z? in p qy V

ry 2rt z 2 ^-|-&c. haec figura affumendo — gz s 7

^-7=^ & ^ migrat in aliam iibi aequalem
cujus Abfciffa x ex data Ordinata v determinatur
per aequationem minus affeftam v a in e fv w -j-gv 2 »

&c. = in k iv« mv 2W dj- &c sV in
p-pqv«^- rv 2 ” &c.

COROL. IX,

Curva omnis cujus Ordinata eft t/ 1 jn
f e ;j; p f I{4„ gz 2 " -H & c - x e -\-1 z" -f- gz^&c.^' 1 x
|a -j- b|ez f -ft fz r +» -|- gz’"!'”~j-&c.| T j ? ft fits = &

affumantur x = ez' -j-fz p +" -j-gz ’'+ 2 °•,

migrat in aliam fibi squalem cujus ordi¬
nata eft x* x a-j-bx'l “• Et nota quod ordinata prior

l l 9$l

In hoc Corollario evadit fimplicior ponendo x — t,
vel ponendo x = i & efficiendo ut radix dignitatis
extrahi poffit cujus index eft », vel etiam ponendo
= i = ut alios cafus praete-

ream.

CO ILO L. X.

Pro ez' -J-r gzTP** &c.

-j-2» g 2 ^ 2 "' 1 "1“ & c - k -J- lz» ~j- mz 2 » -j- &c. & ilz*' 1
‘4" -nnz 2 "' I 4 '&c. fcribantur R, r, S & s refpe£tive, &
Curva omni s cujus ordinata eft xSr 1 Rs in R*- 1 S** -1
x aS v -\- bR x |, ft fite=”“ = " = •, z = a , *-* = .

& R t S? = x, migrat in aliam fibi aequalem cujus or¬
dinata eft x 3 - x a-j-bx^j" Et nota quod Ordinata
prior evadit fimplicior, ponendo unitates pro r, v %
& » vel tfi & faciendo ut radix dignitatis extrahi
poffit cujus index eft *>, vel ponendo * _= —i vel
o .

PROP. X. PROB. III.

Invenire figuras fimpliciffimas cum quibus Curva
quoevis geometrice compari poteft, cujus ordinatim
applicata y per aequationem non affe&am ex data ab-
fcifla z determinatur*

C c c

CAS,

[ m 1

C A S. I.

Sit Ordinata az 8 ' 1 , & area erit jaz 8 , ut ex Prop.V.
ponendob=o=c=d=f~g=h & e~ i, facilecol-
ligitur.

CAS. II.

Sit Ordinata az 8 ' 1 x e -j- fz" j- gz 2 j K " 1 ■-]- & c - & ft
curva cum figuris reftilineis geometrice comparari
poteft, quadrabitur per Prop. V. ponendo b = o = c
■ = d. Sin minus convertetur in aliam eurvam fibi

squalem cujus Ordinata eft ~x~ x e-|-fx-*j-gx l &c.
gen Carol, a. Prop. IX. Deinde fi de dignitatum
indicibus i per Prop. VII. rejiciantur uni¬

te tes donee dignitates ills fiant quam minima:, de-
venietur ad figuras fimpliciffimas qua; hac rations
colligi poffunt. Dein harurn unaquaeq; per Corol.5.
Prop. IX. dat aliam quae nonnunquam llmplicior
eft. Et ex his per Prop. III. & Corol. 9 & 10,
Prop* IX. inter fe collatis, figurse adhuc fimpliciores
quandoq- prodeunt, Deniq; ex figuris fimplicif-
ftmis affumptis- fa£to regreffu computabitur area
qusfita.

CAS.

[ IPS ]

C A S. III.

Sit Ordinata z 9 ' 1 x a bz* cz 2 » -j-- &c.

x e-jp fz« 4"' gz 2M & c -|^ 1 •> & haec figura fi quadrari
poteft, quadrabitur per Prop, V. Sin minus, di~

ftinguenda eft ordinata in partes z 9-1 x axe^fz®

~ ! r gz 2M -J-* Scc-p' 1 , z 9 ' 1 x bz w x e fz« \-g
See. & per Caf. a. invenienda funt figura fimpli-
ciffima cum quibus figura partibus illis refpon-
dentes comparari poflunt. Nam area figurarum
partibus illis refpondentium fub fignis fuis 4* & —
conjun&a component aream totam quafitam.

CAS. IV.

C ) V i

Sit Ordinata z 9 ' 1 x a -fi b z« -fi-cz 2>| ~fi Sec , x
e-J-’fz” gz 2 » -fi &c.H x k -fi lz w ^mz^-fi&c.J^ 1 :
& fi Curva quadrari poreft,quadrabitur per Prop. VI
Sin minus, convertetur in fimpliciorem per Coral 4.
Prop. IX. ac deinde comparabitur cum figuris fim-
pliciffimis per Prop. VIII. & Coral 6, 9 & 10,
Prop a IX, ut fit in Cafu a & 3.

• C A S. V, -

Si Ordinata ex variis partibus conftat, partes
lingula pro ordinatis curvarum totidem liabenda
funt 5 & curva ilia quotquot quadrari poffunt,figilk-

C c c a ‘ tim

c*9<n

tim quadrandse funt, earumq; ordinatae de ordinata
tota demendae. Dein Curva quam ordinatae pars
refidua defignat feorfim ( ut in Cafu a, 3 & 4,)
cumfiguris fimpliciffimis comparanda eft cum qui-
bus comparari poteft. Et fumma arearum omnium
pro area Curra propofitse habenda eft.

CORQL . r.

Hinc etiam Curva .omnis' cujus Ordinata eft ra¬
dix quadratica affefta cequationis fuoe, cum figuris
fimpliciffimis feu re&ilineis feu curvilineis com-
pari poteft. Nam radix ilia ex duabus partibus
femper conftat qux feorfim fpe&atae non funt aequa-
num radices affefifoe. Proponatur sequatio aayy
4- zzyy — 2a ? y ~^ 2 z 3 y — z% & extradta radix erit:
_ a* % lJr av^-^az 3 —z 4 cujus pars rationalis

Ad ZZ

aq-j-z* „ . aVa 4 -j- aaz 3 ’ —

- aa - q r g ' &■ pars irrationahs - ; a ~ fy;-~ - tunt

ordinatae curvarum quae per hanc Propofitionem

vel quadrari poflunt vel cum figuris fimpliciffimis

comparari cum quibus collationem geometricam ad-

mittunt.

’CORQL. II.

Et curva omnis cujus Ordinata per aequationem
quamvis affedtam definitur quae per Corol. 7. Prop.
IX. in aequationem non affectam migrat, vel qua-

dratur

C *97 ]

drattir per hanc Propofitionem fi quadrari poteft vel
comparatur cum figuris fimplicifllmis cum quibus
compari poteft. Et hac ratione Curva omnis quadra-
tur cujus aequatio eft trium terminorum. Nam aequa-
tio ilia ft affetta fit tranlmutatur in non affectum per
Corol.7. Prop.IX. ac deinde per Corol. a & 5. Prop.
IX. in fimplicffimam migrando, dat vel quadratu-
ram figurae fi quadrari poteft, vel curvam fimplicif-
fimam quacum comparatur.

COROL. III.

Et Curva omnis cujus Ordinata per cequationem
quamvis affectum definitur quae per Corol. 8. Prop.
IX. in aequationem quadraticam affe&am migrat ;
vel quadratur per hanc Propofitionem & hujus Co¬
rol. 1. fi quadrari poteft, vel comparatur cum figu¬
ris fimpliciffimis cum quibus collationem geombtri-
ram admittit.

SCHOLIUM.

Ubi quadrandae funt figurae; ad Regulas hafce
generates femper recurrere nimis moleftum effet:
prseftat Figuras quae fimpliciores funt & magis ufui
effe poffunt femel quadrare & quadratures in Ta-
bulam referre, deinde Tabulam confulere quoties
ejufmodi Curvam aliquam quadrare oportet. Hu¬
jus autem generis funt Tabulae duse fequentes, in
quibus z denotat Abfciffam, y Ordinatam rebtan-

gulam

CTQ CfQ

[i 9 8]

ulam & t Aream Curvse quadrands, & d, e, f, g,
. h," funt quantitates datas cum fignis fuis-|- &—.

TABULA

Cur varum fimpliciorum qua quadrari pojfrnt.

Curvarum forms. Curvarum ares.
Forma prima.

dz n ~ l = v.

Forma fecunda.

~Z" = t.

dz^ 1

’ee-|-2efzjj-|-ffz2 Y 1

Forma tertia.

dz w j^—d

»ee |-|jefzjj — VCl

= t.

1 * dzJVe+fz«==y. fjfR 3 = t 5 exiftente R=Ye-J-fz*

a. dz?J Ve= y. ”^1!^ dR 3 — t

3 . d Z ?;v'e+E=y. ■‘—gy'iy

4. dzfj Ve-i-fz" = y.

dR» = t.

3 n

945 nf+

dR’ = t .

Forma quarta,
dz rl

1»

v /e-’ r fz tt

dz 2rI

y«

<v /e-|-fzjj

2 d

7f R — t.

=^r»dR=t.

dz 3 " 1

dz 3 "' 1

dz 4 "' 1

v'c- r tz,i

= y>

[ 199 ]

16ee—<8 efz #- 1 - 6ffz2»

——cIR = t.

-—96e3-|-48eefzM“-3 i 5effz2 l )-]-3of3 zjH
io 5»4

cl R. ~ t .

TABULA

Curvarum Jimpliciornm (pace cum Ettipft &
Hyperbola compart pojfunt.

Sit jam aGD vei PGD vei GDS Seftio
Conica cujus area ad Quadraturam Curvae pro- Fig. 5
polite reqmiritur, fitq; ejus centrum A, Axis K a,
Vertex a, Semiaxis conjugatus AP, datum Abfciffae
principium A vei a vei ^ AbfciiTa AB vei aB vei
*B - x, Ordinata reStangula BD = v 5 & Area
ABDP vei aBDG vei aBDG^s, exiftente «G Or¬
dinata ad pundtum Jungantur KD y AD, aD. Du-
catur Tangens DT occurrens Abfciffe AB in T.

& compleatur parallelogrammum A EDO, Et
fiquando ad quadraturam Curvae propolit^e requri
runtur areas duarum Sefitionein Conicaruuq dica-
tur pofterioris AbfciiTa Ordinata T, 8 c Area ^

Sit autem differentia duarum quantitatum ubi in-
certum eft utrum pofterior de priori an prior de po»
fteriori fubduci debeat.

Curva-

Curvarum Formse. Settionis Conicse. Curvarum Area;,

Forma

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[*°$]

In Tabulis hifce, feries Curvarum cujufq; Forms
utrinq; in infinitum continuari poteft. Scilicet
in Tabula prima, in numeratoribus arearurn for¬
ms tertise & quarts, numeri coefficientes initialium
terminorum (2,—4., 16,—96, 868, &c.) generan-
tur multiplicando numeros—2, —4, —6, —10,&c.
in fe continuo, & fubfequentium terminorum coef¬
ficientes ex initialibus derivantur multiplicando
ipfos gradatim, in Forma quidem tertia, per — J
——I,—f» —To &c. in quarta vero per -f’

~h~~b —l’ & c - Et Denominator-urn coefficientes
3, 15, 105, &c. prodeunt multiplicando numeros
1, 3, 5, 7, 9, &c. infe continuo.

In fecunda vero Tabula, Feries Curvarum forms
prims, fecund®, quints, (ext®, non® & decims ope
folius divifionis, & forms reliqus ope Propofitio-
nis tertis & quarts, utrinq; producuntur in in¬
finitum.

Quinetiam hs feries- mutando lignum numeri
variari folent. Sic enim, e. g. Curva |\/"e_pfz»= y

evadit ■

P R O P. IX. T H E O R. VIII,

'‘Sit A DIC Curva qusvis Abfciffam habens v; a 0 .
AB==z & Ordinatam BD=y, 8 c fit A E K C Curva * 9 ‘
alia cujus Ordinata B E squalis eft prioris ares

ABC

[ 206 ]

ADB ad unitatem applicate, &AFLC Curva
tertia cujus Ordinata B F asqualis eft fecundas areas
AEB ad unitatem applicatse, & AGMC Curva
quarta cujus Ordinata B G squalls eft tertias areas
AFB ad unitatem applicatas, &AHNC Curva
quinta cujus Ordinata BH asqualis eft quartae areas
A G B ad unitatem applicatae, & ftc deinceps in
infinitum. Et funto A, B, C 7 D, E, &c. Areae Cur¬
va rum Ordinatas habentium y, zy 7 z 2 y, z 3 y ? z 4 y ?
& Abrciflfam communem z.

Detur Abfcifla quaevis AC=t, fitq; BC=t—z
& funto P 7 Q, R ? S 5 T areas Curvarum Ordi¬
natas habentium x, xy 7 xxy, x*y, x 4 y & Abfciflam
communem x.

Terminenter autem has areas omnes ad Abfciflam
totam datam A C 7 nec non ad Ordinatam pofitione
datam & infinite produdam CI : & erit arearum
tub initio pofitarum prima ADIC=A=P 5 fecunda
AEKC=tA—B=Q.Tertia AFLC = «£=?£+£ = i R .
Quarta AGMC — ^a— sttB-htc—p _ ig . Q u i' n ta

A H N C —iiA

4tD -|-E __ i -p
24 24 a •

CO-

[ 207 ]

CORO L.

llnde fi Curvs quarum Ordinata* funt y, zy,
z'y r . z 3 y, &c. vel y, xy, x l y, x ! y, &c. quadrari
poffunt, quadrabuntur etiam Curva? ADIC^ AEKC,
AFLC, AGMC ? &c. & habebuntur Ordinatf BE,
BF-, BG 5 BH areis Cur varum proportionates.

SCHOLIUM.

Quantitatum fiuentium fluxiones effe primas 9
fecundas, tertias, quartas 9 aliafq; diximus fiipra.
Hce liuxiones funt ut termini ferierum infinita-
rum convergentium. lie fi z M fit quantitas fluens &

fluendo evadat z-|-o|« 9 deinde refolvatur in feriem
convergentem z M 'j'HOz rI -^~ooz n 3 -^^±^o^z yr3

c. terminus primus hujus feriei z” erit quan¬
titas ilia fluens 9 lecundus «oz rI erit ejus incremen-
tum primum feu differentia prima cui nafeenti pro-
portionalis eft ejus fluxio prima 9 tertius — oz r2
erit ejus inerementum fecundum feu differentia fe-
cunda cui nafeenti proportionalis eft ejus fluxio

fecunda, quartus ~~ 3 ^ o 3 z K ' 3 erit ejus increment

turn tertium feu differentia tertia cui nafeenti
fluxio tertia proportionalis eft. & fic deinceps in
infinitum.

Exponi

[ 208 ]

Exponi autem poffunt hsefluxiones per Curvarum
Ordinatas BD, BE, BE, BG, BH, &c. Ut fi

Ordinata BE (—fit quantitas fluens, erit

ejus fluxio prima ut ordinata B D, Si BF (=^~)

fit quantitas fluens, erit ejus fluxio prima ut Or¬
dinata BE & fluxioiecunda ut Ordinata BD. Si
BH (— fit quantitas fluens, erunt ejus fluxio-
nes, prima, fecunda, tertia & quarta, ut Ordinate
BG, BF, BE, BDrefpeaive.

Et hinc in aequationibus quas quantitates tantum
duas incognitas involvunt, quarum una eft quan¬
titas uniformiter fluens & altera eft fluxio quslibet
quantitatis alterius fluentis, inveniri poteft fluens
ilia altera per quadraturam Curvarum. Exponatur
enina fluxio ejus per Ordinatam BD, & fi hcec fit
fluxio prima, quaeratur area ADB=BExi 5 fi
fluxio iecunda, quaeratur area AEB = BFx i, fi
fluxio tertia, quaeratur area AFB — BGxq&c
& area inventa erit exponens fluentis quaeiitse.

Sed & in aequationibus quae ftuentem & ejus
fluxionem primam fine altera fluente, vel duas
ejufdem fluentis fluxiones, primam & fecundam,
vel fecundam & tertiam, vel tertiam & quartam,
See. fine alter utra fluente involvunt: inveniri pofi
font fluentes per quadraturam Curvarum, Sit

aequatio aav = av A- vv , exiftente v = B E,

v =BD, z =AB & z = i, & aequatio ilia com-

plendo dimenfiones fluxionum, evadet aav = avz

feu jtxt? ==Z. Jam float v uniformiter &

fit

[ 209 ]

fit ejus fluxio v=i & erit-q—= z, & quadrando
Curvam cujus Ordinata eft & Abfcifla v, ha-=

bebitur ftuens z. Adhsec fit aequatio aav=av-j-vv
exiftente v=BF, v=BE, v=BD & z—AB &
per relationem inter v & v feu B D & B E invenie-
tur relatio inter A B & B E ut in exemplo fuperiore.
Deinde per hanc rektionem invenietur relatio in¬
ter A B & BF quadrando Curvam AEB*

iEquationes quae tres incognitas quantitates invol-
vunt aliquando reduci poflunt ad aequationes quae
duas tantum involvunt, & in his cafibus fluentes
invenientur ex fluxionibus ut fupra. Sit aequatio

a—-bx m =cxy«y +dy 2 «yy. Ponatur y«y=v & erit
a—bx ra cxv~}-dvv, Hxc aequatio quadrando Cur¬
vam cujus Abiciffa eft x & Ordinata v dat aream
v 5 & aequatio altera y«y=v regrediendo ad fluentes
dat =v. Unde habetur fluens y.

Quinetiam in aequationibus quse tres incognitas
involvunt & ad aequationes quae duas tantum in-
volvunt reduci non poflunt , fluentes quandoq;
prodeunt p er quadraturam Cur varum. Sit aequatio
ax m -j-bxf — rex r ’ r y s sex r y j w —fy y\ exiftente

x = i.Et pars pofterior rex^ 1 y s -j-sex^y y s ~ l — fy y £ ,
regrediendo ad fluentes, fit exry s —- JL q U ^

proinde e ft ut area Curvae cujus Abfcifla eft x &
Ordinata ax m -] i x l j^ & inde datur fluens y.

E e e

Sit

C 2FO J

__ , 9

Sit aequatio x * ax ra T bx"| p =—err: 1 Et fluens

cujus fluxio eft x xax m ft- bx"| p erit ait area Curv^e
cujus Abfcifla eft x & Ordinata eft a x m ft- bx«l p .

Item fluens cuius fluxio eft dyYrI - erit ut area Curvs

v'e+fyo

cuius Abfcifla eft y & Ordinata - yyi ’— id eft

(per Cafum i- Formae quartae Tab. I.) ut area
Ve-f fy”- Pone ergo^Ve-j-fy" aequalem areae

Curvae cujus Abfcifla eft x & Ordinata ax® -\- b x"^
Sc habebitur fluens y,

Et nota quod fluens omnis quae ex fluxione prima
colligitur augeri poteft vel minui quantitate quavis
non fluente. Qua: ex fluxione fecunda colligitur
augeri poteft vel minui quantitate quavis cujus
fluxio fecunda nulla eft. Quae ex fluxione tertia
colligitur augeri poteft vel minui quantitate quavis
cujus fluxio tertia nulla eft. Et fic deinceps in in¬
finitum,

Poftquam vero- fluentes ex fluxionibus col left?:
£unt ? , fi de veritate Condufionis dubitatur,, fluxio-
nes fluentium inventarum viciffim colligendae funt
Sc cum fluxionibus fub initio propofitis comparand^
Mam fi' prodeunt aequales Conclufio rede fe ha-

bet

[ 21 1 1

Bet: fin minus, corrigenda? funt fiuentes fic, ut
earum fluxiones fiuxionibus fub initio propofitis
aequentur. Nam & Fluens pro lubitu aflfumi po-
teft & affumptio corrigi ponendo fluxionem flu-
entis affumpt® cequalem fluxioni propofitae, & ter-
minos homologos inter fe comparand©*

Et his principiis via ad majora fternitur-

IT**

i\j •■d

ERRATA.

BOOK I. OfOftick*.

P Art i. p.3. 1 . 20 . Properties which, ib-p-5- 1 .$* and that C, p .6. I.9. DE,^ p.2i. I.23.

are two Pays, p.27.1.6. in the Margin put Fig. 14 ££ 15, p.30.1.7. Ma t , I.9, M, p.
44.1.15. aswttfpropofed, 11.52. 1. 17. a paper Circle, p.57. l.uit. emerging, p,6o. I.25.
contain with the, p.64.1.18. and 14th, p.65. i.13. at the, p.66. [^.Semicircular, p.67.
I.25. Center, I.31. ^ Inches, p.68. 1.8. to 16, 1.9. or p.71.1.1. bife& 9 p.72.1.13.

falls, I.20. being. Part II. p.86.1.5. lelopipede, p.89.1.9. made by, p.93. l.i 8. to 77^
1.28,29, by the third Axiom of the firft Pan of this Book, the Laws, p.105. I.5. fee repre-
femei, p. 144.1. 244, b, r«, 5. P- for Lib.i. Ub.l. write

Ttmi.Pam. p.122. 1 .^.iniico, jj.130.la9.ro the Jttgle, p.132. l.S. by tie bright-
vcfs, p.1-3-5.1.14. Ferlf in the, l.i6.firfl Fanyou, p.136. [.26. firfi Part, [.27. lights,
p.137.1.20. green, accordingly an, i^. 138.1. 21. Prop.6. Part. 2. p.139. I.5. on which,
^.i^.l.ij.XTwhichhave been, p.143* 1*7 •purple, \.i6.feveral Lights,1.2^. of white.

BOOK. II.

P.5. 1. 5 .nicely the, p. 7.1.9.^, x denote, I.28. them divers ,p, 10.1. 24. xooo to 1024,
p.ii.l.n. olicpiities, I. p.17. 1 * 4 * Hf P* 25. 1 . 11. 10^ p»3i. !• 12. more com¬
pounded, p.55.1 .3. foes re fie ft, I.24. and therefore their Colours arife, p.65. I.5. cqrpuf^
eles can , p.71.1.17. given breadth, p.84* 1* 4 * a.re tothofe, p. 96. 1. 24. Ohfervation of
this Pan of this Book, p.103.1.17. was to thelhickhefs, p. 105.1.19. of this white Ping,
p.107.1.20. become equal to the third of thofe.

Enumeratio Line arum.

p.143.1.20. datas[ignis fm, p. 144.1.27. refipiciunt, p. 146° 1 .^fmtJfympmo, p.
154.1.13. cx-\-d dat Qrdinatamy =5»1.14* quxgeneratur.

Ouadratura Curvarum.

p.x68.1.24.re^i ^Bjp.r/^.l.ult. J fz«,p.183.1.1 q.a,b,c,tic. e,fig,tic. k,l, m*

‘tic. p. 18$. I.4. in 2,9 -i> p.188.1.14. z9dr «<r, p. 190. 1.1 &yei

192* I.18. gz p *~*~ 2 n* P*i93*i.n. aS^-bRr;

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Full text of "Opticks: or, A treatise of the reflections, refractions, inflexions and colours of light : also two treatises of the species and magnitude of curvilinear figures"

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